Richard Karp: Algorithms and Computational Complexity
嘉宾：Richard Karp
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本篇目录12 章节
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The following is a conversation with Richard Karp,
a professor at Berkeley and one of the most important figures
in the history of theoretical computer science.
In 1985, he received the Turing Award
for his research in the theory of algorithms,
including the development of the admirance Karp algorithm
for solving the max flow problem on networks,
Hopcroft-Karp algorithm for finding
maximum cardinality matchings in bipartite graphs,
and his landmark paper in complexity theory
called "Reducibility Among Combinatorial Problems,"
in which he proved 21 problems to be NP-complete.
This paper was probably the most important catalyst
in the explosion of interest in the study of NP-completeness
and the P versus NP problem in general.
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And now, here's my conversation with Richard Karp.
You wrote that at the age of 13,
you were first exposed to plane geometry
and was wonderstruck by the power
and elegance of formal proofs.
Are there problems, proofs, properties, ideas
in plane geometry that, from that time,
that you remember being mesmerized by
or just enjoying to go through to prove various aspects?
- So Michael Rabin told me this story
about an experience he had when he was a young student
who was tossed out of his classroom for bad behavior
and was wandering through the corridors of his school
and came upon two older students
who were studying the problem
of finding the shortest distance
between two non-overlapping circles.
And Michael thought about it and said,
"You take the straight line between the two centers
"and the segment between the two circles is the shortest
"because a straight line is the shortest distance
"between the two centers.
"And any other line connecting the circles
"would be on a longer line."
And I thought, and he thought,
and I agreed that this was just elegant,
that pure reasoning could come up with such a result.
- Certainly the shortest distance
from the two centers of the circles is a straight line.
Could you once again say what's the next step in that proof?
- Well, any segment joining the two circles,
if you extend it by taking the radius on each side,
you get a path with three edges
which connects the two centers.
And this has to be at least as long as the shortest path,
which is the straight line.
- The straight line, yeah.
Wow, yeah, that's quite simple.
So what is it about that elegance
that you just find compelling?
- Well, just that you could establish a fact
about geometry beyond dispute by pure reasoning.
I also enjoy the challenge
of solving puzzles in plane geometry.
It was much more fun than the earlier mathematics courses
which were mostly about arithmetic operations
and manipulating them.
- Was there something about geometry itself,
the slightly visual component of it,
that you can visualize? - Oh, yes, absolutely.
Although I lacked three-dimensional vision.
I wasn't very good at three-dimensional vision.
- You mean being able to visualize three-dimensional objects?
- Three-dimensional objects or surfaces,
hyperplanes and so on.
So there I didn't have an intuition,
but for example, the fact that the sum of the angles
of a triangle is 180 degrees is proved convincingly,
and it comes as a surprise that that can be done.
- Why is that surprising?
- Well-- - Well, it is a surprising idea,
I suppose.
- Why is that proved difficult?
- It's not.
That's the point.
It's so easy, and yet it's so convincing.
- Do you remember what is the proof
that it adds up to 180?
- You start at a corner and draw a line
parallel to the opposite side,
and that line sort of trisects the angle
between the other two sides,
and you get a half-plane which has to add up
to 180 degrees, and it consists,
and the angles, by the equality of alternate angles,
what's it called?
You get a correspondence between the angles
created along the side of the triangle
and the three angles of the triangle.
- Has geometry had an impact on,
when you look into the future of your work
with combinatorial algorithms,
has it had some kind of impact in terms of,
yeah, being able, the puzzles, the visual aspects
that were first so compelling to you?
- Not Euclidean geometry particularly.
I think I use tools like linear programming
and integer programming a lot,
but those require high-dimensional visualization,
and so I tend to go by the algebraic properties.
- Right, you go by the linear algebra
and not by the visualization.
- Well, the interpretation in terms of, for example,
finding the highest point on a polyhedron,
as in linear programming, is motivating,
but again, I don't have the high-dimensional intuition
that would particularly inform me,
so I sort of lean on the algebra.
- So to linger on that point,
what kind of visualization do you do
when you're trying to think about,
we'll get to combinatorial algorithms,
but just algorithms in general.
- Yeah.
- What's inside your mind when you're thinking
about designing algorithms?
Or even just tackling any mathematical problem?
- Well, I think that usually an algorithm
involves a repetition of some inner loop,
and so I can sort of visualize the distance
from the desired solution as iteratively reducing
until you finally hit the exact solution.
- And try to take steps that get you closer to the--
- Try to take steps that get closer
and having the certainty of converging.
§02Transcript · 10:41–21:48
So it's basically the mechanics of the algorithm
is often very simple,
but especially when you're trying something out
on the computer.
So for example, I did some work
on the traveling salesman problem,
and I could see there was a particular function
that had to be minimized,
and it was fascinating to see the successive approaches
to the optimum.
- You mean, so first of all,
traveling salesman problem is where you have to visit
every city without ever, only once.
- Yeah, that's right.
Find the shortest path through a set of cities.
- Yeah, which is sort of a canonical,
a standard, a really nice problem
that's really hard in computer science.
- Exactly, yes.
- So can you say again what was nice
about being able to think about the objective function there
and maximizing it or minimizing it?
- Well, it's just that as the algorithm proceeded,
you were making progress, continual progress,
and eventually getting to the optimum point.
- So there's two parts, maybe.
Maybe you can correct me,
but first is like getting an intuition
about what the solution would look like,
and or even maybe coming up with a solution,
and two is proving that this thing
is actually going to be pretty good.
What part is harder for you?
Where does the magic happen?
Is it in the first sets of intuitions,
or is it in the messy details of actually showing
that it is going to get to the exact solution
and it's gonna run at a certain complexity?
- Well, the magic is just the fact
that the gap from the optimum decreases monotonically,
and you can see it happening,
and various metrics of what's going on
are improving all along until finally you hit the optimum.
Perhaps later we'll talk about the assignment problem,
and I can illustrate.
- Illustrate a little better.
- Yeah.
- Now zooming out again, as you write,
Don Knuth has called attention to a breed of people
who derive great aesthetic pleasure
from contemplating the structure of computational processes.
So Don calls these folks geeks,
and you write that you remember the moment
you realized you were such a person,
you were shown the Hungarian algorithm
to solve the assignment problem.
So perhaps you can explain what the assignment problem is
and what the Hungarian algorithm is.
- So in the assignment problem,
you have n boys and n girls,
and you are given the desirability of,
or the cost of matching the i-th boy
with the j-th girl for all i and j.
You're given a matrix of numbers,
and you want to find the one-to-one matching
of the boys with the girls,
such that the sum of the associated costs will be minimized.
So the best way to match the boys with the girls
or men with jobs or any two sets.
- Now any possible matching is possible?
Or there's constraints?
- All one-to-one correspondences are permissible.
If there is a connection that is not allowed,
then you can think of it as having an infinite cost.
- I see, yeah.
- So what you do is to depend on the observation
that the identity of the optimal assignment,
or as we call it, the optimal permutation,
is not changed if you subtract a constant
from any row or column of the matrix.
You can see that the comparison
between the different assignments is not changed by that.
Because if you decrease a particular row,
all the elements of a row by some constant,
all solutions decrease by the cost of that,
by an amount equal to that constant.
So the idea of the algorithm is to start
with a matrix of non-negative numbers
and keep subtracting from rows or entire columns
in such a way that you subtract the same constant
from all the elements of that row or column,
while maintaining the property
that all the elements are non-negative.
- Simple.
- Yeah, and so what you have to do
is find small moves which will decrease the total cost
while subtracting constants from rows or columns.
And there's a particular way of doing that
by computing the kind of shortest path
through the elements in the matrix.
And you just keep going in this way
until you finally get a full permutation of zeros
while the matrix is non-negative,
and then you know that that has to be the cheapest.
- Is that as simple as it sounds?
So the shortest path through the matrix part?
- Yeah, the simplicity lies in how you find,
I oversimplified slightly,
you will end up subtracting a constant
from some rows or columns
and adding the same constant back to other rows and columns.
So as not to reduce any of the zero elements,
you leave them unchanged.
But each individual step modifies
several rows and columns by the same amount,
but overall decreases the cost.
- So there's something about that elegance
that made you go, "Aha, this is a beautiful,"
like it's amazing that something like this,
something so simple can solve a problem like this.
- Yeah, it's really cool.
If I had mechanical ability,
I would probably like to do woodworking
or other activities where you sort of shape something
into something beautiful and orderly.
And there's something about the orderly,
systematic nature of that iterative algorithm
that is pleasing to me.
- So what do you think about this idea of geeks,
as Don Knuth calls them?
What do you think, is it something specific
to a mindset that allows you to discover the elegance
in computational processes, or is this all of us?
Can all of us discover this beauty?
Were you born this way? (laughs)
- I think so, I always like to play with numbers.
I used to amuse myself
by multiplying four-digit decimal numbers in my head.
And putting myself to sleep by starting with one
and doubling the number as long as I could go.
And testing my memory,
my ability to retain the information.
- And I also read somewhere that you wrote
that you enjoyed showing off to your friends
by, I believe, multiplying four-digit numbers.
- Right.
- A couple of four-digit numbers.
- Yeah, I had a summer job at a beach resort
outside of Boston.
And the other employee,
I was the barker at a skee-ball game.
- Yeah.
- I used to sit at a microphone saying,
"Come one, come all, come in and play skee-ball.
"Five cents to play, a nickel to win," and so on.
- That's what a barker, I wasn't sure if I should know,
but barker, you're the charming, outgoing person
that's getting people to come in.
- Yeah, well, I wasn't particularly charming,
but I could be very repetitious and loud.
And the other employees were sort of juvenile delinquents
who had no academic bent,
but somehow I found that I could impress them
by performing this mental arithmetic.
- Yeah, there's something to that.
Some of the most popular videos on the internet is,
there's a YouTube channel called Numberphile
that shows off different mathematical ideas.
- I see.
- There's still something really profoundly interesting
to people about math, the beauty of it.
Something, even if they don't understand
the basic concept even being discussed,
there's something compelling to it.
What do you think that is?
Any lessons you drew from your early teen years
when you were showing off to your friends with the numbers?
What is it that attracts us
to the beauty of mathematics, do you think?
The general population, not just the computer scientists
and mathematicians.
- I think that you can do amazing things.
You can test whether large numbers are prime.
You can solve little puzzles
about cannibals and missionaries.
(laughing)
- Yeah.
- And there's a kind of achievement, it's puzzle solving.
And at a higher level, the fact that you can do
this reasoning, that you can prove
in an absolutely ironclad way that some of the angles
of a triangle is 180 degrees.
- Yeah, it's a nice escape from the messiness
of the real world where nothing can be proved.
So, and we'll talk about it, but sometimes the ability
to map the real world into such problems
where you can't prove it is a powerful step.
- Yeah.
- It's amazing that we can do it.
- Of course, another attribute of geeks
§03Transcript · 21:50–32:53
is they're not necessarily endowed
with emotional intelligence.
And so they can live in a world of abstractions
without having to master the complexities
of dealing with people.
- So just to link on the historical note,
as a PhD student in 1955, you joined
the computational lab at Harvard
where Howard Aiken had built the Mark I
and the Mark IV computers.
Just to take a step back into that history,
what were those computers like?
- The Mark IV filled a large room,
much bigger than this large office
that we're talking in now.
And you could walk around inside it.
There were rows of relays.
You could just walk around the interior
and the machine would sometimes fail
because of bugs, which literally meant
flying creatures landing on the switches.
So I never used that machine for any practical purpose.
The lab eventually acquired
one of the earlier commercial computers.
- This is already in the '60s.
- No, in the mid '50s.
- The mid '50s?
- Or the late '50s.
- There was already commercial computers in the--
- Yeah, we had a UNIVAC with 2,000 words of storage.
And so you had to work hard to allocate
the memory properly to also the excess time
from one word to another depended on the number
of the particular words.
And so there was an art to sort of arranging
the storage allocation to make fetching data rapid.
- Were you attracted to this actual physical world
implementation of mathematics?
So it's a mathematical machine that's actually
doing the math physically for you?
- No, not at all.
I think I was attracted to the underlying algorithms.
- But did you draw any inspiration?
So could you have imagined,
like what did you imagine was the future
of these giant computers?
Could you have imagined that 60 years later
we'd have billions of these computers all over the world?
- I couldn't imagine that,
but there was a sense in the laboratory
that this was the wave of the future.
In fact, my mother influenced me.
She told me that data processing was gonna be really big
and I should get into it.
- She's a smart woman.
- Yeah, she was a smart woman.
And there was just a feeling that this was going
to change the world, but I didn't think of it
in terms of personal computing.
I had no anticipation that we would be walking around
with computers in our pockets or anything like that.
- Did you see computers as tools,
as mathematical mechanisms to analyze
sort of theoretical computer science,
or as the AI folks, which is an entire other community
of dreamers, as something that could one day
have human level intelligence?
- Well, AI wasn't very much on my radar.
I did read Turing's paper about the
- The Turing test, computing and intelligence?
- Yeah, the Turing test.
- What'd you think about that paper?
Was that just like science fiction?
- I thought that it wasn't a very good test
because it was too subjective.
So I didn't feel that the Turing test
was really the right way to calibrate
how intelligent an algorithm could be.
- But to linger on that, do you think it's part,
because you've come up with some incredible tests
later on, tests on algorithms, right?
That are strong, reliable, robust
across a bunch of different classes of algorithms.
But returning to this emotional mess that is intelligence,
do you think it's possible to come up with a test
that's as ironclad as some of the
computational complexity work?
- Well, I think the greater question
is whether it's possible to achieve
human level intelligence.
- Right, so that's, so first of all,
let me, at the philosophical level,
do you think it's possible to create algorithms
that reason and would seem to us
to have the same kind of intelligence as human beings?
- It's an open question.
It seems to me that most of the achievements
have operate within a very limited set of ground rules
and for a very limited, precise task,
which is a quite different situation
from the processes that go on in the minds of humans,
which, where they have to sort of function
in changing environments.
They have emotions, they have physical attributes
for exploring their environment.
They have intuition, they have desires, emotions.
And I don't see anything in the current achievements
of what's called AI that come close to that capability.
I don't think there's any computer program
which surpasses a six-month-old child
in terms of comprehension of the world.
- Do you think this complexity of human intelligence,
all the cognitive abilities we have, all the emotion,
do you think that could be reduced one day
or just fundamentally, can it be reduced
to a set of algorithms or an algorithm?
So can a Turing machine achieve human-level intelligence?
- I am doubtful about that.
I guess the argument in favor of it
is that the human brain seems to achieve
what we call intelligence,
cognitive abilities of different kinds.
And if you buy the premise that the human brain
is just an enormous interconnected set of switches,
so to speak, then in principle,
you should be able to diagnose
what that interconnection structure is like,
characterize the individual switches,
and build a simulation outside.
But while that may be true in principle,
that cannot be the way we're eventually
gonna tackle this problem.
That's, you know, that does not seem
like a feasible way to go about it.
So there is, however, an existence proof that
if you believe that the brain is just a network
of neurons operating by rules,
I guess you could say that that's an existence proof
of the ability to build,
the capabilities of a mechanism.
But it would be almost impossible
to acquire the information
unless we got enough insight
into the operation of the brain.
- But there's so much mystery there.
Do you think, what do you make of consciousness,
for example?
There's something, as an example,
something we completely have no clue about,
the fact that we have this subjective experience.
- Right.
- Is it possible that this network of,
this circuit of switches is able to create
something like consciousness?
- To know its own identity.
- Yeah, to know the algorithm, to know itself.
- To know itself.
I think if you try to define that rigorously,
you'd have a lot of trouble.
- Yeah, that seems to.
- So I know that there are many who
believe that general intelligence can be achieved,
and there are even some who feel certain
that the singularity will come
and we will be surpassed by the machines,
which will then learn more and more about themselves
and reduce humans to an inferior breed.
I am doubtful that this will ever be achieved.
- Just for the fun of it,
could you linger on what's your intuition,
why you're doubtful?
So there are quite a few people
that are extremely worried about this existential threat
of artificial intelligence, of us being left behind
by the super intelligent new species.
What's your intuition why that's not quite likely?
- Just because none of the achievements
in speech or robotics or natural language processing
or creation of flexible computer assistants
or any of that comes anywhere near close
to that level of cognition.
- What do you think about ideas of sort of,
if we look at Moore's law, an exponential improvement
to allow us, that would surprise us?
Sort of our intuition fall apart
with exponential improvement because,
I mean, we're not able to kind of,
we kind of think in linear improvement.
- Yeah.
- We're not able to imagine a world
that goes from the Mark I computer to an iPhone 10.
- Yeah.
- So do you think we could be really surprised
by the exponential growth?
Or on the flip side, is it possible
that also intelligence is actually way, way, way, way harder
even with exponential improvement to be able to crack?
- I don't think any constant factor improvement
could change things.
I mean, given our current comprehension
§04Transcript · 32:57–44:31
of what cognition requires,
it seems to me that multiplying the speed of the switches
by a factor of a thousand or a million
will not be useful until we really understand
the organizational principle behind the network of switches.
- Well, let's jump into the network of switches
and talk about combinatorial algorithms if we could.
Let's step back for the very basics.
What are combinatorial algorithms?
And what are some major examples
of problems they aim to solve?
- A combinatorial algorithm is one which deals
with a system of discrete objects
that can occupy various states
or take on various values from a discrete set of values
and need to be arranged or selected in such a way
as to achieve some, to minimize some cost function
or to prove the existence
of some combinatorial configuration.
So an example would be coloring the vertices of a graph.
- What's a graph?
Let's step back.
So what, it's fun to ask one of the greatest
computer scientists of all time
the most basic questions in the beginning of most books.
But for people who might not know,
but in general how you think about it, what is a graph?
- A graph, that's simple.
It's a set of points, certain pairs of which
are joined by lines called edges.
And they sort of represent the,
in different applications represent the interconnections
between discrete objects.
So they could be the interactions,
interconnections between switches in a digital circuit
or interconnections indicating the communication patterns
of a human community.
- And they could be directed or undirected.
And then as you've mentioned before,
you might have costs.
- Right, they can be directed or undirected.
They can be, you can think of them as,
if you think if a graph were representing
a communication network,
then the edge could be undirected,
meaning that information could flow along it
in both directions, or it could be directed
with only one way communication.
A road system is another example of a graph
with weights on the edges.
And then a lot of problems of optimizing
the efficiency of such networks or learning about
the performance of such networks
are the object of combinatorial algorithms.
So it could be scheduling classes at a school
where the vertices, the nodes of the network
are the individual classes and the edges indicate
the constraints which say that certain classes
cannot take place at the same time,
or certain teachers are available only
for certain classes, et cetera.
Or I talked earlier about the assignment problem
of matching the boys with the girls,
where you have a, there a graph with an edge
from each boy to each girl
with a weight indicating the cost.
Or in logical design of computers,
you might want to find a set of so-called gates,
switches that perform logical functions,
which can be interconnected to realize some function.
So you might ask how many gates do you need
in order to, for a circuit to give a yes output
if at least a given number of its inputs are ones
and no if fewer are present.
- My favorite is probably all the work with network flows.
So anytime you have, I don't know why it's so compelling,
but there's something just beautiful about it.
It seems like there's so many applications
and communication networks and traffic flow
that you can map into these.
And then you can think of pipes and water
going through pipes and you can optimize it
in different ways.
There's something always visually
and intellectually compelling to me about it.
And of course you've done work there.
- Yeah, so there the edges represent channels
along which some commodity can flow.
It might be gas, it might be water,
it might be information.
- Maybe supply chain as well, like products being--
- Products flowing from one operation to another.
- Yeah.
- And the edges have a capacity,
which is the rate at which the commodity can flow.
And a central problem is to determine,
given a network of these channels,
in this case, the edges are communication channels.
The challenge is to find the maximum rate
at which the information can flow along these channels
to get from a source to a destination.
And that's a fundamental combinatorial problem
that I've worked on.
Jointly with the scientist, Jack Edmonds,
we, I think we're the first to give a formal proof
that this maximum flow problem through a network
can be solved in polynomial time.
- Which I remember the first time I learned that,
just learning that in maybe even grad school,
I don't think it was even undergrad, no.
Algorithm, yeah.
Do network flows get taught in basic algorithms courses?
- Yes, probably.
- Okay, so yeah, I remember being very surprised
that max flow is a polynomial time algorithm.
- Yeah.
- That there's a nice, fast algorithm that solves max flow.
But, so there is an algorithm named after you in Edmonds.
The Edmond-Karp algorithm for max flow.
So what was it like tackling that problem
and trying to arrive at a polynomial time solution?
And maybe you can describe the algorithm,
maybe you can describe what's the running time complexity
that you showed.
- Yeah, well, first of all,
what is a polynomial time algorithm?
- Yeah.
- Perhaps we could discuss that.
- So yeah, let's actually just even, yeah.
That's what is algorithmic complexity?
What are the major classes of algorithm complexity?
So we, in a problem like the assignment problem
or scheduling schools or any of these applications,
you have a set of input data,
which might, for example, be a set of vertices
connected by edges with,
you're given for each edge, the capacity of the edge.
And you have algorithms,
which are, think of them as computer programs
with operations such as addition, subtraction,
multiplication, division, comparison of numbers and so on.
And you're trying to construct an algorithm
based on those operations,
which will determine in a minimum number
of computational steps, the answer to the problem.
In this case, the computational step
is one of those operations.
And the answer to the problem is, let's say,
the configuration of the network
that carries the maximum amount of flow.
And an algorithm is said to run in polynomial time
if as a function of the size of the input,
the number of vertices, the number of edges and so on.
The number of basic computational steps
grows only as some fixed power of that size.
A linear algorithm would execute a number of steps
linearly proportional to the size.
Quadratic algorithm would be steps proportional
to the square of the size and so on.
And algorithms whose running time is bounded
by some fixed power of the size
are called polynomial algorithms.
- And that's supposed to be
relatively fast class of algorithms.
- That's right.
Theoreticians take that to be the definition
of an algorithm being efficient.
And we're interested in which problems can be solved
by such efficient algorithms.
One can argue whether that's the right definition
of efficient because you could have an algorithm
whose running time is the 10,000th power
of the size of the input,
and that wouldn't be really efficient.
- And in practice, it's oftentimes reducing
from an N squared algorithm to an N log N
or a linear time is practically the jump
that you wanna make to allow a real world system
to solve a problem.
- Yeah, that's also true because especially
as we get very large networks,
the size can be in the millions
and then anything above N log N,
where N is the size,
would be too much for a practical solution.
- Okay, so that's polynomial time algorithms.
What other classes of algorithms are there?
So that usually they designate polynomials
as a letter P.
- Yeah.
- There's also NP, NP complete, NP hard.
- Yeah.
- So can you try to disentangle those
by trying to define them simply?
- Right, so a polynomial time algorithm
is one whose running time is bounded
by a polynomial in the size of the input.
Then the class of such algorithms is called P.
- In the worst case, by the way, we should say, right?
- Yeah.
§05Transcript · 44:32–56:29
- So for every case of the problem.
- Yes, that's right, and that's very important
that in this theory,
when we measure the complexity of an algorithm,
we really measure the number of steps,
the growth of the number of steps in the worst case.
So you may have an algorithm
that runs very rapidly in most cases,
but if there's any case
where it gets into a very long computation,
that would increase the computational complexity
by this measure.
And that's a very important issue
because there, as we may discuss later,
there are some very important algorithms
which don't have a good standing
from the point of view of their worst case performance
and yet are very effective.
So theoreticians are interested in P,
the class of problems solvable in polynomial time.
Then there's NP, which is the class of problems
which may be hard to solve,
but where the, where when confronted with a solution,
you can check it in polynomial time.
Let me give you an example there.
So if we look at the assignment problem,
so you have n boys, you have n girls,
the number of numbers that you need to write down
to specify the problem instances, n squared.
And the question is
how many steps are needed to solve it.
And Jack Edmonds and I were the first to show
that it could be done in time n cubed.
Earlier algorithms required n to the fourth.
So as a polynomial function of the size of the input,
this is a fast algorithm.
Now to illustrate the class NP,
the question is how long would it take to verify
that a solution is optimal?
So for example, if the input was a graph,
we might want to find the largest clique in the graph,
or a clique is a set of vertices such that any vertex,
each vertex in the set is adjacent to each of the others.
So the clique is a complete subgraph.
- Yeah, so if it's a Facebook social network,
everybody's friends with everybody else,
close clique of friends.
- No, that would be what's called a complete graph,
it would be.
- No, I mean within that clique.
- Within that clique, yeah.
They're all friends.
- So a complete graph is when--
- Everybody is friendly.
- Is everybody is friends with everybody, yeah.
- So the problem might be to determine
whether in a given graph there exists a clique
of a certain size.
- Well, that turns out to be a very hard problem,
but if somebody hands you a clique
and asks you to check whether it is,
hands you a set of vertices and asks you to check
whether it's a clique, you could do that simply
by exhaustively looking at all of the edges
between the vertices in the clique
and verifying that they're all there.
- And that's a polynomial time algorithm.
- That's a polynomial, so the problem of finding the clique
appears to be extremely hard,
but the problem of verifying a clique
to see if it reaches a target number of vertices
is easy to verify.
So finding the clique is hard, checking it is easy.
Problems of that nature are called
non-deterministic polynomial time algorithms,
and that's the class NP.
- And what about NP complete and NP hard?
- Okay, let's talk about problems
where you're getting a yes or no answer
rather than a numerical value.
So either there is a perfect matching
of the boys with the girls or there isn't.
It's clear that every problem in NP is also in NP.
If you can solve the problem exactly,
then you can certainly verify the solution.
On the other hand, there are problems in the class NP.
This is the class of problems that are easy to check,
although they may be hard to solve.
It's not at all clear that problems in NP lie in P.
So for example, if we're looking at scheduling classes
at a school, the fact that you can verify
when handed a schedule for the school,
whether it meets all the requirements,
that doesn't mean that you can find the schedule rapidly.
So intuitively, NP, non-deterministic polynomial,
checking rather than finding,
is going to be harder than,
is going to include, is easier.
Checking is easier, and therefore the class of problems
that can be checked appears to be much larger
than the class of problems that can be solved.
- And then you keep adding appears to
and sort of these additional words
that designate that we don't know for sure yet.
- We don't know for sure.
So the theoretical question,
which is considered to be the most central problem
in theoretical computer science,
or at least computational complexity theory,
combinatorial algorithm theory,
question is whether P is equal to NP.
If P were equal to NP, it would be amazing.
It would mean that every problem
where a solution can be rapidly checked
can actually be solved in polynomial time.
We don't really believe that's true.
If you're scheduling classes at a school,
we expect that if somebody hands you a satisfying schedule,
you can verify that it works.
That doesn't mean that you should be able
to find such a schedule.
So intuitively, NP encompasses a lot more problems than P.
- So can we take a small tangent
and break apart that intuition?
So do you, first of all, think that the biggest
sort of open problem in computer science,
maybe mathematics, is whether P equals NP?
Do you think P equals NP?
Or do you think P is not equal to NP?
If you had to bet all your money on it.
- I would bet that P is unequal to NP,
simply because there are problems
that have been around for centuries
and have been studied intensively in mathematics,
and even more so in the last 50 years
since the P versus NP was stated.
And no polynomial time algorithms have been found
for these easy to check problems.
So one example is a problem that goes back
to the mathematician Gauss,
who was interested in factoring large numbers.
So we know what a number is prime
if it cannot be written as the product
of two or more numbers unequal to one.
So if we can factor a number like 91, it's seven times 13.
But if I give you 20 digit or 30 digit numbers,
you're probably going to be at a loss
to have any idea whether they can be factored.
So the problem of factoring very large numbers
does not appear to have an efficient solution.
But once you have found the factors,
expressed the number as a product of two smaller numbers,
you can quickly verify that they are factors of the number.
- And your intuition is a lot of people finding,
you know, a lot of brilliant people
have tried to find algorithms.
This one particular problem, there's many others like it
that are really well studied,
and it would be great to find an efficient algorithm for.
- Right, and in fact, we have some results
that I was instrumental in obtaining,
following up on work by the mathematician Stephen Cook,
to show that within the class NP,
of easy to check problems,
there's a huge number that are equivalent in the sense
that either all of them or none of them lie in P.
And this happens only if P is equal to NP.
So if P is unequal to NP, we would also know
that virtually all the standard combinatorial problems,
if P is unequal to NP,
none of them can be solved in polynomial time.
- Can you explain how that's possible
to tie together so many problems in a nice bunch
that if one is proven to be efficient, then all are?
- The first and most important stage of progress
was a result by Stephen Cook,
who showed that a certain problem
called the satisfiability problem of propositional logic
is as hard as any problem in the class P.
So the propositional logic problem is expressed
in terms of expressions involving the logical operations
and or and not operating on variables
that can be either true or false.
So an instance of the problem would be some formula
involving and or and not.
And the question would be whether there is an assignment
of truth values to the variables in the problem
that would make the formula true.
So for example, if I take the formula A or B
and A or not B and not A or B and not A or not B
and take the conjunction of all four
of those so-called expressions,
you can determine that no assignment of truth values
to the variables A and B will allow that conjunction
of what are called clauses to be true.
So that's an example of a formula in propositional logic
involving expressions based on the operations and or and not.
That's an example of a problem which is not satisfiable.
There is no solution that satisfies
all of those constraints.
§06Transcript · 56:31–1:08:21
- And that's like one of the cleanest
and fundamental problems in computer science.
It's like a nice statement of a really hard problem.
- It's a nice statement of a really hard problem.
And what Cook showed is that every problem in NP
can be re-expressed as an instance
of the satisfiability problem.
So to do that, he used the observation
that a very simple abstract machine
called the Turing machine can be used
to describe any algorithm.
An algorithm for any realistic computer
can be translated into an equivalent algorithm
on one of these Turing machines,
which are extremely simple.
- So Turing machine, there's a tape
and you can walk along that tape.
- Yeah, you have data on a tape
and you have basic instructions,
a finite list of instructions,
which say if you're reading a particular symbol on the tape
and you're in a particular state,
then you can move to a different state
and change the state of the number,
the element that you were looking at,
the cell of the tape that you were looking at.
- And that was like a metaphor and a mathematical construct
that Turing put together
to represent all possible computation.
- All possible computation.
Now, one of these so-called Turing machines
is too simple to be useful in practice,
but for theoretical purposes,
we can depend on the fact that an algorithm
for any computer can be translated
into one that would run on a Turing machine.
And then using that fact,
he could sort of describe
any possible non-deterministic polynomial time algorithm,
any algorithm for a problem in NP
could be expressed as a sequence of moves
of the Turing machine described
in terms of reading a symbol on the tape
while you're in a given state
and moving to a new state
and leaving behind a new symbol.
And given that fact that any
non-deterministic polynomial time algorithm
can be described by a list of such instructions,
you could translate the problem
into the language of the satisfiability problem.
- Is that amazing to you, by the way,
if you take yourself back when you were first thinking
about this space of problems?
How amazing is that?
- It's astonishing.
When you look at Cook's proof,
it's not too difficult to sort of figure out
why this is so,
but the implications are staggering.
It tells us that this, of all the problems in NP,
all the problems where solutions are easy to check,
they can all be rewritten in terms of
the satisfiability problem.
- Yeah, it's adding so much more weight
to the P equals NP question,
because all it takes is to show that one--
- That's right.
- One algorithm in this class--
- So the P versus NP can be re-expressed
as simply asking whether the satisfiability problem
of propositional logic is solvable in polynomial time.
But there's more.
I encountered Cook's paper
when he published it in a conference in 1971.
Yeah, so when I saw Cook's paper
and saw this reduction of each of the problems in NP
by a uniform method to the satisfiability problem
of propositional logic,
that meant that the satisfiability problem
was a universal combinatorial problem.
And it occurred to me through experience I had had
in trying to solve other combinatorial problems
that there were many other problems
which seemed to have that universal structure.
And so I began looking for reductions
from the satisfiability to other problems.
One of the other problems would be
the so-called integer programming problem
of determining whether there's a solution
to a set of linear inequalities
involving integer variables.
- Just like linear programming,
but there's a constraint
that the variables must remain integers.
- Integers, in fact, must be either zero or one.
It could only take on those values.
- And that makes the problem much harder.
- Yes, that makes the problem much harder.
And it was not difficult to show
that the satisfiability problem can be restated
as an integer programming problem.
- So can you pause on that?
Was that one of the first problem mappings
that you tried to do?
And how hard is that map?
And you said it wasn't hard to show,
but that's a big leap.
- It is a big leap, yeah.
Well, let me give you another example.
Another problem in NP
is whether a graph contains a clique of a given size.
And now the question is,
can we reduce the propositional logic problem
to the problem of whether there's a clique
of a certain size?
Well, if you look at the propositional logic problem,
it can be expressed as a number of clauses,
each of which is of the form A or B or C,
where A is either one of the variables in the problem
or the negation of one of the variables.
And an instance of the propositional logic problem
can be rewritten using operations of Boolean logic.
Can be rewritten as the conjunction of a set of clauses,
the and of a set of ors,
where each clause is a disjunction,
an or of variables or negated variables.
So the question of,
in the satisfiability problem,
is whether those clauses can be simultaneously satisfied.
Now to satisfy all those clauses,
you have to find one of the terms in each clause,
which is going to be true in your truth assignment,
but you can't make the same variable both true and false.
So if you have the variable A in one clause
and you want to satisfy that clause by making A true,
you can't also make the complement of A true
in some other clause.
- And so the goal is to make every single clause true
if it's possible to satisfy this.
And the way you make it true is at least...
- One term in the clause must be true.
- Got it.
So now we, to convert this problem
to something called the independent set problem,
where you're just sort of asking for a set of vertices
in a graph such that no two of them are adjacent,
sort of the opposite of the clique problem.
So we've seen that we can now express that as...
Finding a set of terms, one in each clause,
without picking both the variable
and the negation of that variable.
Because if the variable is assigned the truth value,
the negated variable has to have the opposite truth value.
- Right.
- And so we can construct a graph
where the vertices are the terms in all of the clauses.
And you have an edge between two terms,
if an edge between two occurrences of terms.
Either if they're both in the same clause,
because you're only picking one element from each clause.
And also an edge between them
if they represent opposite values of the same variable,
because you can't make a variable both true and false.
And so you get a graph
where you have all of these occurrences of variables.
You have edges, which mean that you're not allowed
to choose both of the variables.
Both ends of the edge,
either because they're in the same clause
or they're negations of one another.
- Right, and that's a, first of all,
sort of to zoom out, that's a really powerful idea
that you could take a graph
and connect it to a logic equation somehow.
And do that mapping for all possible formulations
of a particular problem on a graph.
- Yeah.
- I mean, that still is hard for me to believe
that that's possible.
Like, what do you make of that,
that there's such a union of,
there's such a friendship among all these problems across
that somehow are akin to combinatorial algorithms,
that they're all somehow related.
I know it can be proven, but what do you make of it,
that that's true?
- Well, that they just have the same expressive power.
You can take any one of them
and translate it into the terms of the other.
- The fact that they have the same expressive power
also somehow means that they can be translatable.
- Right.
And what I did in the 1971 paper
was to take 21 fundamental problems,
commonly occurring problems of packing,
covering, matching, and so forth,
lying in the class NP,
and show that the satisfiability problem
can be re-expressed as any of those,
§07Transcript · 1:08:24–1:19:40
that any of those have the same expressive power.
So--
- And that was like throwing down the gauntlet
of saying there's probably many more problems like this.
- Right.
- But that's just saying that,
look, that they're all the same.
- They're all the same, but not exactly.
Yeah, they're all the same in terms of
whether they are rich enough to express any of the others.
But that doesn't mean that they have
the same computational complexity.
But what we can say is that either all of these problems
or none of them are solvable in polynomial time.
- Yeah, so where does NP completeness and NP hard
as classes fit?
- Oh, that's just a small technicality.
So when we're talking about decision problems,
that means that the answer is just yes or no.
There is a clique of size 15
or there's not a clique of size 15.
On the other hand, an optimization problem would be asking,
find the largest clique.
The answer would not be yes or no,
it would be 15.
So when you're asking for the,
when you're putting a valuation on the different solutions
and you're asking for the one with the highest valuation,
that's an optimization problem.
And there's a very close affinity
between the two kinds of problems.
But the counterpart of being the hardest decision problem,
the hardest yes/no problem,
the counterpart of that is to minimize
or maximize an objective function.
And so a problem that's hardest in the class
when viewed in terms of optimization,
those are called NP hard rather than NP complete.
- And NP complete is for decision problems.
- And NP complete is for decision problems.
- So if somebody shows that P equals NP,
what do you think that proof will look like
if you were to put on yourself,
if it's possible to show that as a proof
or to demonstrate an algorithm?
- All I can say is that it will involve concepts
that we do not now have and approaches that we don't have.
- Do you think those concepts are out there
in terms of inside complexity theory,
inside of computational analysis of algorithms?
Do you think there's concepts
that are totally outside of the box
that we haven't considered yet?
- I think that if there is a proof that P is equal to NP
or that P is not equal to NP,
it'll depend on concepts that are now outside the box.
- Now if that's shown, either way, P equals NP or P not,
well, actually P equals NP,
what impact, you kind of mentioned a little bit,
but can you linger on it?
What kind of impact would it have
on theoretical computer science
and perhaps software systems in general?
- Well, I think it would have enormous impact
on the world in either way case.
If P is unequal to NP, which is what we expect,
then we know that for the great majority
of the combinatorial problems that come up,
since they're known to be NP complete,
we're not going to be able to solve them
by efficient algorithms.
However, there's a little bit of hope
in that it may be that we can solve most instances.
All we know is that if a problem is not in P,
then it can't be solved efficiently on all instances.
But basically, it will,
if we find that P is unequal to NP,
it will mean that we can't expect always
to get the optimal solutions to these problems.
And we have to depend on heuristics
that perhaps work most of the time
or give us good approximate solutions.
- So we would turn our eye towards the heuristics
with a little bit more acceptance and comfort on our hearts.
- Exactly.
- Okay, so let me ask a romanticized question.
What to you is one of the most
or the most beautiful combinatorial algorithm
in your own life or just in general
in the field that you've ever come across
or have developed yourself?
- I like the stable matching problem
or the stable marriage problem very much.
- What's the stable matching problem?
- Yeah.
Imagine that you want to marry off N boys
with N girls.
And each boy has an ordered list
of his preferences among the girls,
his first choice, his second choice,
through her Nth choice.
And each girl also has an ordering of the boys,
his first choice, second choice, and so on.
And we'll say that a matching,
a one-to-one matching of the boys with the girls is stable
if there are no two couples in the matching
such that the boy in the first couple
prefers the girl in the second couple to her mate
and she prefers the boy to her current mate.
In other words, if there is, the matching is stable
if there is no pair who want to run away with each other,
leaving their partners behind.
- Gotcha. (laughs)
Yeah.
- Actually, this is relevant to matching residents
with hospitals and some other real-life problems,
although not quite in the form that I described.
So it turns out that there is,
that for any set of preferences, a stable matching exists.
And moreover, it can be computed by a simple algorithm
in which each boy starts making proposals to girls.
And if a girl receives a proposal,
she accepts it tentatively, but she can drop it if,
she can end it, she can drop it later
if she gets a better proposal from her point of view.
And the boys start going down their list,
proposing to their first, second, third choices
until stopping when a proposal is accepted.
But the girls, meanwhile, are watching the proposals
that are coming into them,
and the girl will drop her current partner
if she gets a better proposal.
- And the boys never go back through the list?
- They never go back, yeah.
So once they've been denied.
(both laugh)
- They don't try again?
- They don't try again,
because the girls are always improving their status
as they receive better and better proposals.
The boys are going down their list,
starting with their top preferences.
And one can prove that the process will come to an end
where everybody will get matched with somebody,
and you won't have any pair that want to abscond
from each other.
- Do you find the proof or the algorithm itself beautiful,
or is it the fact that with the simplicity
of just the two marching,
I mean, the simplicity of the underlying rule
of the algorithm, is that the beautiful part?
- Both, I would say.
And you also have the observation that you might ask,
who is better off, the boys who are doing the proposing
or the girls who are reacting to proposals?
And it turns out that it's the boys who are doing the best,
and as each boy is doing at least as well
as he could do in any other staple matching.
So there's a sort of lesson for the boys
that you should go out and be proactive
and make those proposals, go for broke.
- I don't know if this is directly mappable
philosophically to our society,
but certainly seems like a compelling notion.
And like you said, there's probably a lot
of actual real-world problems that this could be mapped to.
- Yeah, well, you get complications.
For example, what happens when a husband and wife
want to be assigned to the same hospital?
So you have to take those constraints into account.
And then the problem becomes NP-hard.
- Why is it a problem for the husband and wife
to be assigned to the same hospital?
- No, it's desirable.
- Desirable.
- Or at least go to the same city.
So you can't, if you're--
- I see.
- If you're assigning residents to hospitals.
- And then you have some preferences
for the husband and wife or for the hospitals.
- The residents have their own preferences.
References, residents, both male and female,
have their own preferences.
The hospitals have their preferences.
But if resident A, the boy, is going to Philadelphia,
then you'd like his wife also to be assigned to a hospital.
- To be assigned to a hospital in Philadelphia.
- Which step makes it a NP-hard problem that you mentioned?
- The fact that you have this additional constraint.
That it's not just the preferences of individuals,
but the fact that the two partners to a marriage
have to be assigned to the same place.
- I'm being a little dense.
The perfect matching?
No, not the perfect, stable matching is what you refer to.
That's when two partners are trying to--
- Okay, what's confusing you is that in the first
interpretation of the problem,
I had boys matching with girls.
§08Transcript · 1:19:42–1:32:39
- Yes.
- In the second interpretation,
you have humans matching with institutions.
- With institutions.
And there's a coupling between within the,
gotcha, within the humans.
- Any added little constraint will make it
an NP-hard problem.
- Well, yeah.
- Okay.
By the way, the algalene you mentioned,
was it one of yours?
- No, no, that was due to Gale and Shapley.
My friend David Gale passed away before he could get part
of the Nobel Prize, but his partner Shapley shared
in the Nobel Prize with somebody else for--
- Economics?
- For economics, for ideas stemming
from the stable matching idea.
- So you've also developed yourself some elegant,
beautiful algorithms.
Again, picking your children, so the Robin Karp algorithm
for string searching, pattern matching, Edmund Karp
algorithm for max flows we mentioned,
Hopcroft Karp algorithm for finding maximum cardinality
matchings in bipartite graphs.
Is there ones that stand out to you,
ones you're most proud of, or just whether it's beauty,
elegance, or just being the right discovery,
development in your life that you're especially proud of?
- I like the Robin Karp algorithm because it illustrates
the power of randomization.
So the problem there is to decide whether a given long string
of symbols from some alphabet contains a given word,
whether a particular word occurs within some very much
longer word, and so the idea of the algorithm
is to associate with the word that we're looking for,
a fingerprint, some number or some combinatorial object
that describes that word, and then to look for an occurrence
of that same fingerprint as you slide along the longer word.
And what we do is we associate with each word a number.
So we, first of all, we think of the letters that occur
in a word as the digits of, let's say, decimal
or whatever base here, whatever number of different symbols
there are in the alphabet.
- That's the base of the numbers, yeah.
- Right, so every word can then be thought of as a number
with the letters being the digits of that number.
And then we pick a random prime number in a certain range
and we take that word viewed as a number
and take the remainder on dividing that number by the prime.
- So coming up with a nice hash function.
- It's a kind of hash function.
- Yeah, it gives you a little shortcut
for that particular word.
- Yeah, so that's the--
- It's very different than other algorithms of its kind
that were trying to do search, string matching.
- Yeah, which usually are combinatorial
and don't involve the idea of taking a random fingerprint.
- Yes.
- And doing the fingerprinting has two advantages.
One is that as we slide along the long word,
digit by digit, we keep a window of a certain size,
the size of the word we're looking for.
And we compute the fingerprint of every stretch
of that length and it turns out that just a couple
of arithmetical operations will take you
from the fingerprint of one part to what you get
when you slide over by one position.
So the computation of all the fingerprints is simple.
And secondly, it's unlikely if the prime is chosen randomly
from a certain range that you will get two of the segments
in question having the same fingerprint.
And so there's a small probability of error
which can be checked after the fact
and also the ease of doing the computation
because you're working with these fingerprints
which are remainder's modulo some big prime.
- So that's the magical thing about randomized algorithms
is that if you add a little bit of randomness,
it somehow allows you to take a pretty naive approach,
a simple looking approach
and allow it to run extremely well.
So can you maybe take a step back and say,
like what is a randomized algorithm,
this category of algorithms?
- Well, it's just the ability to draw a random number
from some range or to associate a random number
with some object or to draw at random from some set.
So another example is very simple
if we're conducting a presidential election
and we would like to pick the winner.
In principle, we could draw a random sample
of all of the voters in the country.
And if it was of substantial size, say a few thousand
then the most popular candidate in that group
would be very likely to be the correct choice
that would come out of counting all the millions of votes.
And of course we can't do this
because first of all,
everybody has to feel that his or her vote counted.
And secondly, we can't really do a purely random sample
from that population.
And I guess thirdly, there could be a tie in which case
we wouldn't have a significant difference
between two candidates.
- But those things aside,
if you didn't have all that messiness of human beings,
you could prove that that kind of random picking would--
- You guess that random picking would solve the problem
with a very low probability of error.
Another example is testing whether a number is prime.
So if I want to test whether 17 is prime,
I could pick any number between one and 17
and raise it to the 16th power modulo 17
and you should get back the original number.
That's a famous formula due to Fermat
about it's called Fermat's little theorem
that if you take any number A in the range
zero through N minus one
and raise it to the N minus one power modulo N,
you'll get back the number A if A is prime.
So if you don't get back the number A,
that's a proof that a number is not prime.
- Wow. (laughs)
- And you can show that suitably define
the probability that you will get a value unequal,
you will get a violation of Fermat's result is very high
and so this gives you a way of rapidly proving
that a number is not prime.
It's a little more complicated than that
because there are certain values of N
where something a little more elaborate has to be done
but that's the basic idea.
Taking an identity that holds for primes
and therefore if it ever fails on any instance
for a non-prime, you know that the number is not prime.
It's a quick choice, a fast choice,
fast proof that a number is not prime.
- Can you maybe elaborate a little bit more
of what's your intuition why randomness works so well
and results in such simple algorithms?
- Well, the example of conducting an election
where you could take in theory, you could take a sample
and depend on the validity of the sample
to really represent the whole,
is it just the basic fact of statistics
which gives a lot of opportunities.
And I actually exploited that sort of random sampling idea
in designing an algorithm for counting the number
of solutions that satisfy a particular formula
and propositional logic.
- A particular, so some version
of the satisfiability problem or?
- A version of the satisfiability problem.
- Is there some interesting insight
that you wanna elaborate on?
Like what some aspect of that algorithm
that might be useful to describe?
- So you have a collection of formulas
and you want to count the number of solutions
that satisfy at least one of the formulas.
And you can count the number of solutions
that satisfy any particular one of the formulas,
but you have to account for the fact
that that solution might be counted many times
if it solves more than one of the formulas.
And so what you do is you sample from the formulas
according to the number of solutions
that satisfy each individual one.
And that way you draw a random solution,
but then you correct by looking at the number of formulas
that satisfy that random solution and don't double count.
So if you, you can think of it this way.
So you have a matrix of zeros and ones,
and you wanna know how many columns
of that matrix contain at least one one.
And you can count in each row how many ones there are.
So what you can do is draw from the rows
according to the number of ones.
If a row has more ones, it gets drawn more frequently.
But then if you draw from that row,
you have to go up the column and looking at
where that same one is repeated in different rows
and only count it as a success or a hit
if it's the earliest row that contains the one.
- Right.
- And that gives you a robust statistical estimate
of the total number of columns
that contain at least one of the ones.
So that is an example of the same principle
that was used in studying random sampling.
Another viewpoint is that if you have a phenomenon
that occurs almost all the time,
then if you sample one of the occasions where it occurs,
you're most likely to, and you're looking for an occurrence,
a random occurrence is likely to work.
So that comes up in solving identities,
solving algebraic identities.
§09Transcript · 1:32:42–1:43:22
You get two formulas that may look very different.
You wanna know if they're really identical.
What you can do is just pick a random value
and evaluate the formulas at that value
and seeing if they agree.
And you depend on the fact
that if the formulas are distinct,
then they're gonna disagree a lot.
And so therefore a random choice
will exhibit the disagreement.
And if there are many ways for the two to disagree,
and you only need to find one disagreement,
then random choice is likely to yield it.
- And in general,
so we've just talked about randomized algorithms,
but we can look at the probabilistic analysis
of algorithms.
And that gives us an opportunity to step back.
And as we've said,
everything we've been talking about is worst case analysis.
- Right.
- Could you maybe comment on the usefulness
and the power of worst case analysis
versus best case analysis, average case, probabilistic?
How do we think about the future
of theoretical computer science, computer science,
in the kind of analysis we do of algorithms?
Does worst case analysis still have a place,
an important place,
or do we want to try to move forward
towards kind of average case analysis?
- Yeah.
- And what are the challenges there?
- So if worst case analysis shows
that an algorithm is always good, that's fine.
If worst case analysis is used to show
that the problem, that the solution is not always good,
then you have to step back and do something else to ask,
how often will you get a good solution?
- Just to pause on that for a second,
that's so beautifully put,
because I think we tend to judge algorithms.
We throw them in the trash the moment
their worst case is shown to be bad.
- Right.
And that's unfortunate.
I think a good example is going back
to the satisfiability problem.
There are very powerful programs called SAT solvers,
which in practice fairly reliably solve instances
with many millions of variables that arise
in a digital design or in proving programs correct
in other applications.
And so in many application areas,
even though satisfiability,
as we've already discussed, is NP-complete,
the SAT solvers will work so well
that the people in that discipline tend to think
of satisfiability as an easy problem.
So in other words, just for some reason
that we don't entirely understand,
the instances that people formulate
in designing digital circuits or other applications
are such that satisfiability is not hard to check.
And even searching for a satisfying solution
can be done efficiently in practice.
And there are many examples.
For example, we talked about
the traveling salesman problem.
So just to refresh our memories,
the problem is you've got a set of cities,
you have pairwise distances between cities,
and you want to find a tour through all the cities
that minimizes the total cost of all the edges traversed,
all the trips between cities.
The problem is NP-hard,
but people using integer programming codes
together with some other mathematical tricks
can solve geometric instances of the problem
where the cities are, let's say, points in the plane,
and get optimal solutions to problems
with tens of thousands of cities.
Actually, it'll take a few computer months
to solve a problem of that size,
but for problems of size 1,000 or 2,
it'll rapidly get optimal solutions,
provably optimal solutions,
even though, again, we know that it's unlikely
that the traveling salesman problem
can be solved in polynomial time.
- Are there methodologies,
like rigorous systematic methodologies for,
you said in practice.
In practice, this algorithm is pretty good.
Are there systematic ways of saying,
in practice, this one is pretty good?
So in other words, average case analysis,
or you've also mentioned that average case
kind of requires you to understand
what the typical case is, typical instances,
and that might be really difficult.
- That's very difficult.
So after I did my original work
on getting, showing all these problems to be NP-complete,
I looked around for a way to get some,
shed some positive light on combinatorial algorithms,
and what I tried to do was to study
problems, behavior on the average or with high probability,
but I had to make some assumptions
about what's the probability space,
what's the sample space,
what do we mean by typical problems?
That's very hard to say,
so I took the easy way out
and made some very simplistic assumptions.
So I assumed, for example,
that if we were generating a graph
with a certain number of vertices and edges,
then we would generate the graph
by simply choosing one edge at a time at random
until we got the right number of edges.
That's a particular model of random graphs
that has been studied mathematically a lot,
and within that model,
I could prove all kinds of wonderful things,
I and others who also worked on this.
So we could show that we know exactly how many edges
there have to be in order for
there be a so-called Hamiltonian circuit
that's a cycle that visits each vertex exactly once.
We know that if the number of edges
is a little bit more than n log n,
where n is the number of vertices,
then such a cycle is very likely to exist,
and we can give a heuristic
that'll find it with high probability.
And we got the community in which I was working,
got a lot of results along these lines,
but the field tended to be rather lukewarm
about accepting these results as meaningful
because we were making such a simplistic assumption
about the kinds of graphs that we would be dealing with.
So we could show all kinds of wonderful things,
it was a great playground, I enjoyed doing it,
but after a while, I concluded that
it didn't have a lot of bite
in terms of the practical application.
- Okay, so there's too much into the world of toy problems.
- Yeah.
- Okay, but all right, is there a way to find
nice representative real-world impactful instances
of a problem on which demonstrate that an algorithm is good?
So this is kind of like the machine learning world,
that's kind of what they at its best tries to do
is find a dataset from the real world
and show the performance,
all the conferences are all focused
on beating the performance on that real-world dataset.
Is there an equivalent in the complexity analysis?
- Not really.
Don Knuth started to collect examples of graphs
coming from various places,
so he would have a whole zoo of different graphs
that he could choose from
and he could study the performance of algorithms
on different types of graphs.
- But there it's really important and compelling
to be able to define a class of graphs.
The actual act of defining a class of graphs
that you're interested in,
it seems to be a non-trivial step
if we're talking about instances
that we should care about in the real world.
- Yeah, there's nothing available there
that would be analogous to the training set
for supervised learning,
where you sort of assume that the world
has given you a bunch of examples to work with.
We don't really have that for problems,
for combinatorial problems on graphs and networks.
- You know, there's been a huge growth,
a big growth of data sets available.
Do you think some aspect of theoretical computer science,
I might be contradicting my own question while saying it,
but will there be some aspect,
an empirical aspect of theoretical computer science
which will allow the fact that these data sets are huge,
we'll start using them for analysis?
- If you want to say something about a graph algorithm,
you might take a social network like Facebook
and looking at subgraphs of that
and prove something about the Facebook graph
and at the same time be respected
in the theoretical computer science community.
- That hasn't been achieved yet, I'm afraid.
- Is that P equals NP, is that impossible?
Is it impossible to publish a successful paper
in the theoretical computer science community
that shows some performance on a real world data set?
Or is that really just those are two different worlds?
§10Transcript · 1:43:26–1:55:29
- They haven't really come together.
I would say that there is a field
of experimental algorithmics where people,
sometimes they're given some family of examples.
Sometimes they just generate them at random
and they report on performance,
but there's no convincing evidence
that the sample is representative of anything at all.
- So let me ask in terms of breakthroughs and open problems,
what are the most compelling open problems to you
and what possible breakthroughs do you see in the near term
in terms of theoretical computer science?
- Well, there are all kinds of relationships
among complexity classes that can be studied.
Just to mention one thing,
I wrote a paper with Richard Lipton in 1979
where we asked the following question.
If you take a problem, a combinatorial problem in NP,
let's say, and you choose,
and you pick the size of the problem,
say it's a traveling salesman problem, but of size 52.
And you ask, could you get an efficient,
a small Boolean circuit tailored for that size, 52,
where you could feed the edges of the graph
in as Boolean inputs and get as an output,
the question of whether or not
there's a tour of a certain length.
And that would, in other words,
briefly what you would say in that case
is that the problem has small circuits,
polynomial size circuits.
Now we know that if P is equal to NP,
then in fact, these problems will have small circuits,
but what about the converse?
Could a problem have small circuits,
meaning that an algorithm tailored
to any particular size could work well
and yet not be a polynomial time algorithm?
That is, you couldn't write it
as a single uniform algorithm good for all sizes.
- Just to clarify, small circuits
for a problem of particular size
or even further constraint,
small circuit for a particular--
- No, for all the inputs of that size.
- Of that size.
Is that a trivial problem for a particular instance?
So coming up, an automated way of coming up with a circuit,
I guess that's just--
- That would be hard, yeah.
But there's the existential question.
Everybody talks nowadays about existential questions,
existential challenges.
You could ask the question,
does the Hamiltonian circuit problem
have a small circuit for every size?
For each size, a different small circuit?
In other words, could you tailor solutions
depending on the size and get polynomial size?
- Even if P is not equal to NP.
- Right.
- That would be fascinating if that's true.
- Yeah, what we proved is that if that were possible,
then something strange would happen in complexity theory.
Some high-level class, which I could briefly describe,
something strange would happen.
So I'll take a stab at describing what I mean by that.
- Let's go there.
- So we have to define this hierarchy
in which the first level of the hierarchy is P
and the second level is NP.
And what is NP?
NP involves statements of the form,
there exists a something such that something holds.
So for example,
there exists a coloring such that a graph can be colored
with only that number of colors.
Or there exists a Hamiltonian circuit.
- There's a statement about this graph.
- Yeah.
And NP deals with statements of that kind,
that there exists a solution.
Now you could imagine a more complicated expression,
which says for all X, there exists a Y
such that some proposition holds involving both X and Y.
So that would say, for example, in game theory,
for all strategies for the first player,
there exists a strategy for the second player
such that the first player wins.
That would be at the second level of the hierarchy.
The third level would be, there exists an A
such that for all B, there exists a C that something holds.
And you can imagine going higher and higher in the hierarchy
and you'd expect that the complexity classes
that correspond to those different cases
would get bigger and bigger.
- What do you mean by bigger and bigger?
- Sorry, they'd get harder and harder to solve.
- Harder and harder, right.
- Harder and harder to solve.
And what Lyfton and I showed was that
if NP had small circuits,
then this hierarchy would collapse down to the second level.
In other words, you wouldn't get any more mileage
by complicating your expressions with three quantifiers
or four quantifiers or any number.
- I'm not sure what to make of that exactly.
- Well, I think it would be evidence
that NP doesn't have small circuits
'cause something so bizarre would happen.
But again, it's only evidence, not proof.
- Well, yeah, that's not even evidence
because you're saying P's not equal to NP
because something bizarre has to happen.
I mean, that's proof by the lack of bizarreness
in our science, but it seems like
just the very notion of P equals NP would be bizarre.
So any way you arrive at, there's no way.
You have to fight the dragon at some point.
- Yeah, okay, well, anyway.
For whatever it's worth, that's what we proved.
- Awesome, so that's a potential space
of open, interesting problems.
Let me ask you about this other world
of machine learning, of deep learning.
What's your thoughts on the history
and the current progress of machine learning field
that's often progressed sort of separately
as a space of ideas and space of people
than the theoretical computer science
or just even computer science world?
- Yeah, it's really very different
from the theoretical computer science world
because the results about it,
algorithmic performance tend to be empirical.
It's more akin to the world of SAT solvers
where we observe that for formulas arising in practice,
the solver does well.
So it's of that type.
It's where we're moving into the empirical
evaluation of algorithms.
Now, it's clear that there've been huge successes
in image processing, robotics,
natural language processing, a little less so,
but across the spectrum of game playing is another one.
There've been great successes.
And one of those effects is that it's not too hard
to become a millionaire if you can get a reputation
in machine learning and there'll be all kinds
of companies that will be willing to offer you the moon
because they think that if they have AI at their disposal,
then they can solve all kinds of problems.
But there are limitations.
One is that the solutions that you get
by from to supervise learning problems
through convolutional neural networks
seem to perform amazingly well,
even for inputs that are outside the training set.
But we don't have any theoretical understanding
of why that's true.
Secondly, the solutions, the networks that you get
are very hard to understand.
And so very little insight comes out.
So yeah, yeah, they may seem to work on your training set
and you may be able to discover whether your photos occur
in a different sample of inputs or not,
but we don't really know what's going on.
We don't know the features that distinguish the photographs
or the objects are not easy to characterize.
- Well, it's interesting 'cause you mentioned
coming up with a small circuit
to solve a particular size problem.
It seems that neural networks are kind of small circuits.
- In a way, yeah.
- But they're not programs.
Sort of like the things you've designed are algorithms,
programs, algorithms.
Neural networks aren't able to develop algorithms
to solve a problem.
It's more of a function.
- They are algorithms, it's just that they're--
- But sort of, yeah, it could be a semantic question,
but there's not a algorithmic style
manipulation of the input.
Perhaps you could argue there is.
- Yeah, well--
- It feels a lot more like a function of the input.
- It's a function, it's a computable function.
Once you have the network, you can simulate it
on a given input and figure out the output.
But if you're trying to recognize images,
then you don't know what features of the image
are really being determined by the input.
Determinant of what the circuit is doing.
The circuit is sort of very intricate,
and it's not clear that the simple characteristics
that you're looking for, the edges of the objects
or whatever they may be, they're not emerging
from the structure of the circuit.
- Well, it's not clear to us humans,
§11Transcript · 1:55:31–2:06:52
but it's clear to the circuit.
- Yeah, well, right.
I mean, it's not clear to sort of the elephant
how the human brain works, but it's clear to us humans,
we can explain to each other our reasoning,
and that's why the cognitive science
and psychology field exists.
Maybe the whole thing of being explainable to humans
is a little bit overrated.
- Oh, maybe, yeah.
I guess you can say the same thing about our brain,
that when we perform acts of cognition,
we have no idea how we do it, really.
- That's true.
- We do, though.
I mean, we, at least for the visual system,
the auditory system, and so on,
we do get some understanding of the principles
that they operate under, but for many deeper
cognitive tasks, we don't have that.
- That's right.
Let me ask.
- Yeah.
- You've also been doing work on bioinformatics.
Does it amaze you that the fundamental building blocks,
so if we take a step back and look at us humans,
the building blocks used by evolution
to build us intelligent human beings
is all contained there in our DNA?
- It's amazing, and what's really amazing
is that we are beginning to learn how to edit,
how to edit DNA, which is very, very fascinating,
this ability to take a sequence,
find it in the genome, and do something to it.
- I mean, that's really taking our biological system
towards the worlds of algorithms.
- Yeah, but it raises a lot of questions.
You have to distinguish between doing it on an individual
or doing it on somebody's germline,
which means that all of their descendants will be affected.
- So that's like an ethical...
- Yeah, so it raises very severe ethical questions,
and even doing it on individuals.
Um, is, there's a lot of hubris involved
that you can assume that knocking out a particular gene
is going to be beneficial because you don't know
what the side effects are going to be.
So we have this wonderful new world of gene editing,
which is very, very impressive,
and it could be used in agriculture,
it could be used in medicine in various ways,
but very serious ethical problems arise.
- What are, to you, the most interesting places
where algorithms, sort of the ethical side,
is an exceptionally challenging thing
that I think we're going to have to tackle
with all of genetic engineering.
But on the algorithmic side,
there's a lot of benefit that's possible.
So is there areas where you see exciting possibilities
for algorithms to help model,
optimize, study biological systems?
- Yeah, I mean, we can certainly analyze genomic data
to figure out which genes are operative in the cell
and under what conditions
and which proteins affect one another,
which proteins physically interact.
We can sequence proteins and modify them.
- Is there some aspect of that
that's a computer science problem,
or is that still fundamentally a biology problem?
- Well, it's a big data,
it's a statistical big data problem for sure.
So, you know, the biological data sets
are increasing our ability to study our ancestry,
to study the tendencies towards disease,
to personalize treatment according to what's in our genomes
and what tendencies for disease we have,
to be able to predict what troubles might come upon us
in the future and anticipate them,
to understand whether you,
for a woman, whether her proclivity for breast cancer
is so strong enough that she would want to take action
to avoid it.
- You dedicate your 1985 Turing Award lecture
to the memory of your father.
- Mm-hmm.
- What's your fondest memory of your dad?
- Seeing him standing in front of a class
at the blackboard, drawing perfect circles by hand,
and showing his ability to attract the interest
of the motley collection of eighth grade students
that he was teaching.
- When did you get a chance to see him
draw the perfect circles?
- On rare occasions, I would get a chance
to sneak into his classroom and observe it.
And I think he was at his best in the classroom.
I think he really came to life
and had fun not only teaching,
but engaging in chit-chat with the students
and ingratiating himself with the students.
And what I inherited from that
is a great desire to be a teacher.
I retired recently and a lot of my former students came,
students with whom I had done research
or who had read my papers or who had been in my classes.
And when they talked about me,
they talked not about my 1979 paper or my 1992 paper,
but about what came away in my classes.
And not just the details, but just the approach
and the manner of teaching.
And so I sort of take pride in the,
at least in my early years as a faculty member at Berkeley,
I was exemplary in preparing my lectures
and I always came in prepared to the teeth
and able therefore to deviate
according to what happened in the class
and to really provide a model
for the students.
- So is there advice you can give out
for others on how to be a good teacher?
So preparation is one thing you've mentioned,
being exceptionally well prepared,
but are there other things,
pieces of advice that you can impart?
- Well, the top three would be preparation,
preparation, and preparation.
(laughing)
- Why is preparation so important, I guess?
- It's because it gives you the ease
to deal with any situation that comes up in the classroom.
And if you discover that you're not getting through
one way, you can do it another way.
If the students have questions,
you can handle the questions.
- Ultimately, you're also feeling the crowd,
the students of what they're struggling with,
what they're picking up,
just looking at them through the questions,
but even just through their eyes.
- Yeah, that's right.
- So the preparation, you can dance.
- You can dance, you can say it another way,
give it another angle.
- Are there, in particular, ideas and algorithms
of computer science that you find
were big aha moments for students,
where they, for some reason, once they got it,
it clicked for them and they fell in love
with computer science?
Or is it individual, is it different for everybody?
- It's different for everybody.
You have to work differently with students.
Some of them just don't need much influence.
They're just running with what they're doing
and they just need an ear now and then.
Others need a little prodding.
Others need to be persuaded to collaborate
among themselves rather than working alone.
They have their personal ups and downs.
So you have to deal with each student as a human being
and bring out the best.
- Humans are complicated.
- Yeah.
- Perhaps a silly question.
If you could relive a moment in your life outside of family
because it made you truly happy,
or perhaps because it changed the direction of your life
in a profound way, what moment would you pick?
- I was kind of a lazy student as an undergraduate
and even in my first year in graduate school.
And I think it was when I started doing research.
I had a couple of summer jobs where I was able to contribute
and I had an idea.
And then there was one particular course
on mathematical methods and operations research
where I just gobbled up the material
and I scored 20 points higher than anybody else in the class
then came to the attention of the faculty.
And it made me realize that I had some ability
that was going somewhere.
- You realize you're pretty good at this thing.
I don't think there's a better way to end it, Richard.
It was a huge honor.
Thank you for decades of incredible work.
Thank you for talking to me.
- Thank you, it's been a great pleasure.
You're a superb interviewer.
(laughing)
- I'll stop it.
Thanks for listening to this conversation with Richard Karp
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§12Transcript · 2:06:55–2:07:26
It really is the best way to support this journey I'm on.
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And now let me leave you with some words from Isaac Asimov.
I do not fear computers.
I fear lack of them.
Thank you for listening and hope to see you next time.
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