P vs NP: what is efficient computation? Great Theoretical Ideas In Computer Science Anupam GuptaCS 15-251 Fall 2005 Lecture 29Dec 8, 2005Carnegie Mellon

P vs NP:what is efficient computation?

Great Theoretical Ideas In Computer Science

Anupam Gupta

CS 15-251 Fall 2005

Lecture 29 Dec 8, 2005 Carnegie Mellon University

A Graph Named “Gadget”

NODES or VERTICES

EDGES

K-COLORING

We define a k-coloring of a graph:

1. Each node get colored with one color2. At most k different colors are used3. If two nodes have an edge between them,

then they must have different colors

A graph is called k-colorable iff it has a k-coloring

Sometimes also called “proper” colorings.

A 2-CRAYOLA Question!

Is Gadget 2-colorable?



No, it contains a triangle.









Yes.

A 3-CRAYOLA Question.

A 3-CRAYOLA Question.

3-Coloring Is Decidable by brute force

Try out all 3n colorings until you determine if G has a 3-coloring.

Membership in 3COLOR is not undecidable.

But is it efficient to decide this?

What is an efficient algorithm?

Is an O(n) time algorithm efficient?

How about O(n log n)?O(n2) ?O(n10) ?O(nlog n) ?O(2n) ?O(n!) ?

O(222n) ?

polynomial time

O(nc) for some constant c

non-polynomialtime

Does an algorithmrunning in O(n100) time

count as efficient?

We consider non-polynomial time algorithms

to be inefficient.

And hence a necessary condition for an algorithm

to be efficient is that it should run in poly-time.

Asking for a poly-time algorithm for a problem

sets a (very) low bar when asking for efficient

algorithms.

The question is: can we achieve even this?

I see!

Once we know that our favorite problems have

polynomial time algorithms, we can then worry about

making them run in O(n log n)

or O(n2) time!

But we don’t know that yet for many common problems…

The class P defined in the 50’s.

Paths, Trees and Flowers,

Jack Edmonds, 1965.

The Intrinsic Computational Difficulty of Functions, Alan Cobham, 1964.

not the correct Alan Cobham this is indeed Jack Edmonds

Paths, Trees and Flowers, Jack Edmonds, 1965.

An explanation is due on the use of the words "efficient algorithm"…I am not prepared to set up the machinery necessary to give it formal meaning, nor is the present context appropriate for doing this…For practical purposes the difference between algebraic and exponential order is more crucial than the difference between [computable and not computable]… It would be unfortunate for any rigid criterion to inhibit the practical development of algorithms which are either not known or known not to conform nicely to the criterion… However, if only to motivate the search for good, practical algorithms, it is important to realize that it is mathematically sensible even to question their existence.

Edmonds called them “good algorithms”

The Intrinsic Computational Difficulty of Functions, Alan Cobham, 1964.

For several reasons the class P seems a natural one to consider. For one thing, if we formalize the definition relative to various general classes of computing machines we seem always to end up with the same well-defined class of functions. Thus we can give a mathematical characterization of P having some confidence it characterizes correctly our informally defined class. This class then turns out to have several natural closure properties, being closed in particular under explicit transformation, composition and limited recursion on notation (digit-by-digit recursion).

if p( ) and q( ) are polynomials, then p(q( )) is also a polynomial

The class P

Definition:

We say a language L µ Σ* is in P if there is a program A and a polynomial p() such that for any x in Σ*, A (given x as input) runs for ≤ p(|x|) time and answers question “is x in L?” correctly.

The class P

Definition:

We say function F: Σ*Σ* is in P if there is

a program A and a polynomial p() such that for any x in Σ*, A (given x as input) runs for ≤ p(|x|) time and A(x) = F(x).

technically called FP, but we will blur the distinction for this lecture

The class P

The set of all languages L that can be recognized in polynomial time.

The set of functions that can be computed in polynomial time.

Why are we looking onlyat languages Σ*?

What if we want to work with graphs or boolean

formulas?

Requiring that L Σ* is not really restrictive, since we can encode graphs and Boolean formulas as strings of 0’s and 1’s.

In fact, we do this all the time: inputs for all our programs are just sequences of 0’s and 1’s encoded in some suitable format.

Languages/functions in P?

Example 1: CONN = {graph G: G is a connected graph}

Algorithm A1:

If G has n nodes, then run depth first search from any node, and count number of distinct nodes you see. If you see n nodes, G CONN, else not.

Time: p1(|x|) = Θ(|x|).


Example 2: 2COLOR = {connected G : vertices of G can be 2-colored so that adjacent nodes don’t have same color} (Such colorings are called “proper colorings”.)

Program A2: Pick a vertex and color it red. Repeat { if uncolored node has both red and blue neighbors, abort. if uncolored node has some neighbors red, color it blue. if uncolored node has some neighbors blue, color it red. } If aborted, then G not 2-colorable, else have 2-coloring.


Example 3: 3COLOR = {G : vertices of G can be 3-colored}

NOT KNOWN!


And now a problem for Boolean circuits C:


And, now a problem dealing with Combinational Circuits:

AND, OR, NOT, 0, 1 gates wired together with no feedback allowed.

x3x2x1

OR

AND

AND

AND

OR

OR

OR

CIRCUIT-SATISFIABILITY

Given a circuit with n-inputs and one output, is there a way to assign 0-1 values to the input wires so that the output value is 1 (true)?

AND

AND

NOT

011

1

Yes, this circuit is Yes, this circuit is satisfiablesatisfiable. It has . It has satisfying satisfying assignmentassignment 110. 110.

Example 4:CIRCUIT-SATISFIABILITY

Given: A circuit with n-inputs and one output, is there a way to assign 0-1 values to the input wires so that the output value is 1 (true)?

“Brute force”: try all 2^n assignments. Exponential time…


NOT KNOWN!

Onto the new class, NP

Recall the class P

We say a language L µ Σ* is in P if there is a program A and a polynomial p() such that for any x in Σ*, A (given x as input) runs for ≤ p(|x|) time and answers question “is x in L?” correctly.can think of A as “proving” that x in L.

The new class NP

We say a language L µ Σ* is in NP if there is a program A and a polynomial p() such that for any x in Σ*,

If x L, there exists a “proof” y with |y| ≤ p(|x|) A(x, y) runs for ≤ p(|x|) time and answers question “x in L” correctly.

If x L, for all “proofs” y A(x, y) answers “x not in L” correctly.

a short proof that x in L

that can be quickly “verified”

Verifier rejects all “fake” proofs

The class NP

The set of languages L for which there exist “short” proofs of membership

(of polynomial length)

that can “quickly” verified (in polynomial time).

Recall: A doesn’t have to find these proofs y; it just needs to be able to verify that y is a “correct” proof.

Which languagesare in NP?

P NP

For any L in P, we can just take y to be the empty string and satisfy the requirements.

Hence, every language in P is also in NP.

$106 question: is NP P ?

Languages/functions in NP?

Example 3: 3COLOR = {G : vertices of G can be 3-colored so}

Short proof y: a color Cv for each node v of G.

Verifier A: For each edge (u,v) in G, ensure Cu Cv

Ensure only 3 colors were used.

Proof length: a list of n colors.Verifier runs in time O(number of edges + number of nodes).

YES

Languages/functions in NP?

Example 4:CIRCUIT-SAT

Given: A circuit with n-inputs and one output, is there a way to assign 0-1 values to the input wires so that the output value is 1 (true)?

Proof: satisfying assignmentVerifier just evaluates circuit.

YES

Summary: P versus NP

Language L is in P if membership in L can be decided in poly-time.

Language L is in NP if each x in L has a short “proof of membership” that can be verified in poly-time.

Fact: P NPQuestion: Does NP P ?

Hamilton Cycle

b

a

e

c

d

f

hi

g

a cycle that passes through every node exactly once

NP contains lots of problemswe don’t know to be in P

Classroom SchedulingPacking objects into binsScheduling jobs on machinesFinding cheap tours visiting a subset of citiesAllocating variables to registersFinding good packet routings in networksDecryption…

Hence proving P = NP would break cryptosystems

The $1,000,000 question

In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven Prize Problems. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million allocated to each.

Problem #4: The P vs NP problem.

How can we prove thatNP P?

I would have to show thatevery language in NP has apolynomial time algorithm…

How do I do that?It may take forever!

Also, what if I forgot one of the languages in NP?

Relax, Bonzo!

We can describe one language L in NP,

such that if this language L is in P,

then NP P.

It is a language that cancapture all other languages

in NP.

The Magic Language: CIRCUITSAT

Example 6: CSAT = {circuit C: C has at least 1 satisfying assignment}

If we can write a program A that takes as input a circuit C A(C) runs in time polynomial(|C|) correctly decides if C is satisfiablethen NP P

Steven Cook Leonid Levin

Theorem [Cook/Levin]:CIRCUITSAT is one language in NP, such that if we can show CIRCUITSAT is in P, then we

have shown NP P.

CIRCUITSAT is a language in NP that can capture all other

languages in NP.

We say CIRCUITSAT is NP-complete.

Proof not difficult, but we defer the proof to 15-451 (Algos) or 15-453 (FLAC)

AND

AND

NOT

3-colorability Circuit Satisfiability

CIRCUIT-SAT and 3COLOR

Two problems that seem quite different, but are substantially the same

A Graph Named “Gadget”

T F

XY

Output

T FX Y

Output

X Y

F F F

F T T

T F T

T T T

T FX Y

Output

X Y OR

F F F

F T T

T F T

T T T

OR

OR

NOT

x y z

xy

z

OR

OR

NOT

x y z

xy

z

How do we force the graph to be 3 colorable exactly when the circuit is satifiable?

OR

OR

NOT

x y z

xy

z

Satisfiability of this circuit = 3-colorability of this graph

TRUETRUE

Let C be an n-input circuit.

GIVEN:3-colorOracle

C

BUILD:SAT

Oracle

Graph composed of gadgets that mimic the gates in C with

one edge from output to false.

You can quickly transform a method to decide 3-coloring into

a method to decide circuit satifiability!

Any language in NP

SAT

can be reduced (in polytime to)an instance of

hence SAT is NP-complete

Any language in NP

SAT

3COLOR



Any language in NP

3COLOR



Hence:

SAT

hence 3COLOR is NP-complete

To show a language L is NP-complete

To show L is NP-complete, we must

a) Show L is in NPb) Reduce all problems in NP to L

or

b) Reduce some NP-complete language (e.g. C-SAT) to L(by exhibiting a function F such that

C is satisfiable F(C) is in L)

NP-complete problems

3COLOR, SAT, bin-packing, classroom scheduling, many other problems NP-complete. proofs follow above pattern.

NP-complete problems: “Hardest” problems in NP (if any of these problems in P, then all NP in P.)

NP-complete problems come up all the time Many problems of practical importance are NP-complete If P NP, we will not have poly-time algorithms for them.

Right now, we don’t know.

Reference

Computers and Intractablity:A guide to the Theory of NP-completeness

by Mike Garey and David Johnson

References

The NP-completeness Column: 1981-2005by David Johnsonhttp://www.research.att.com/~dsj/columns/

The P versus NP problem Official Description, Clay Instituteby Steve Cookhttp://www.claymath.org/millennium/P_vs_NP/

http://www.research.att.com/~dsj/columns/

http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf



Documents

P vs NP: what is efficient computation? Great Theoretical Ideas In Computer Science Anupam GuptaCS 15-251 Fall 2005 Lecture 29Dec 8, 2005Carnegie Mellon