חישוביות וסיבוכיותComputability and

Complexity

Lecture 8

Outline

Last week:• Non-deterministic TM• Poly-time verifiability• The classes NP and coNP

Today:• Poly-time reducibility• NP completeness• Existence of NP-complete problems

P, NP and coNP

• P = class of languages where membership can be decided in polynomial time.

• NP = class of languages where membership can be verified in polynomial time.

• coNP = class of languages whose complement is in NP.

Poly-time Verifiability

Definition: A verifier for a language A is an algorithm V, where

A = {w | string c s.t. V accepts <w,c> }

• V’s running time is measured in terms of |w|• A is polynomially verifiable if it has a poly-time

V

Two Definitions of NP

Definition 1: NP is the class of languages that have a polynomial time verifier.

Definition 2: NP is the class of languages that are decidable by a non-deterministic poly-time TM.

Theorem: Definition 1 is equivalent to Definition 2.

Example: A problem in NP

A path in a directed graph G that goes through each node of G exactly once.

HAMPATH={<G,s,t>|G is a directed graph with a Hamiltonian path from s to t}

c

a

d

b

s

te

Example: A Problem in NP II

• A clique in an undirected graph is a subgraph for which every two nodes are connected.

CLIQUE={<G,k>|G is undirected graph with a k-clique}

4-clique

2-clique

P = NP?

PNP

NP=P

?

NP vs. coNP?

PNP coNP

NP-Completeness

Cook-Levin (early 70’s): Certain problems in NP are at least as hard as any problem in NP.

Such problems are called NP-complete.

Many important problems are NP-complete (e.g. HAMPATH, CLIQUE, SAT, SUBSETSUM).

Most problems in NP: either in P or NP-complete (however some are not known to be neither).

Theoretical Importance

To answer the P vs. NP problem: • Focus on an NP complete problem:• To show P ≠ NP, prove it requires more than poly

time.• To show P = NP, find a poly-time algorithm for it.

P

NP NP=PNPC

P

NPC

Practical Importance

I can’t You’re firedSolve it in poly- time

I.

II.Solve it in poly- time

No one knows to do it. It is NP-hard!

First Example: SAT

Boolean variables: {0,1}Boolean operations: {, , }

0 0 = 0 0 0 = 0 0 = 1

0 1 = 0 1 0 = 1 1 = 0

1 0 = 0 0 1 = 1

1 1 = 1 1 1 = 1

Boolean formula: = (x y) (x z)

The satisfiability (SAT) problem:Given a Boolean expression on n variables, can we

assign values such that the expression is TRUE?

The Cook-Levin Theorem

SAT={<>| is a satisfiable Boolean formula}

Claim: SAT ∈ NP

Cook-Levin theorem: SAT ∈ P if and only if P = NP

Polynomial-Time Reducibility

Definition: A language A is polynomial time mapping reducible to a language B, A p B, if there exists a polynomial-time computable function f: where for every w,

A f() B

f(

f

f

Theorem: If A P B and BP, then AP.

A polynomial algorithm to decide A: • Given w, Compute f(w) in polynomial time• Decide whether f(w) B in polynomial time

A B

f(

f

A f() B

Corollary: If A P B and A P then B P.

A B

f(

f

A f() B

3SAT

SAT={<>| is a satisfiable Boolean formula}

A Boolean formula is in conjunctive normal form (CNF) if it is an AND of clauses, each of which is an OR of literals• Example: (x1 x2) (x1 x3 x4) (x5)

3CNF: each clause has exactly 3 distinct literals• Example: (x1 x2 x3) (x1 x2 x3) (x1 x3 x4)

3SAT={<>| is a satisfiable 3CNF formula}

3SAT PCLIQUE

Theorem: 3SAT is poly-time reducible to CLIQUE.

Proof: Given a 3CNF formula F, build a graph G and specify a number k such that a clique of size k exists in G iff the formula F is satisfiable.

3SAT PCLIQUE

The reduction:• Let F = C1 C2 … Ck be a 3-CNF formula with k clauses,

each of which has 3 distinct literals.

• For each clause, Ci, put three vertices in the graph, one for each literal.

• Put an edge between two vertices if they are in different triples and their literals are consistent, meaning not each other’s negation.

In other words: There are edges between all nodes of G except for nodes in the same triplet and nodes with contradictory labels.

Construction by Examplez)yx(z)yx(z)y(xF

-x y z

-zz

y

x

-y

-x

G

clauseliteral

An edge means ‘these two literals do not contradict each other’.

Construction by Examplez)yx(z)yx(z)y(xF

-x y z

-zz

y

x

-y

-x

1z0,y1,x

G

Any clique of size k must include exactly one literal from each clause.

General Construction

k 3

iji=1 j=1

F = a ij 1 1 n na {x ,¬x ,…,x ,¬x }where

G =(V,E) where

ijV ={a :1 i k,1 j 3}

k is the number of clauses

literals

E = {(aij, ai’,j’): ii’ and aij ai’j’}

The Reduction Argument

We need to show:1. F satisfiable implies G has a clique of size k.

Given a satisfying assignment for F, for each clause pick a literal that is satisfied. Those literals in the graph G form a k-clique.

2. G has a clique of size k implies F is satisfiable.Given a k-clique in G, assign TRUE to each literal in the

clique. This yields a satisfying assignment to F (why?).

Example: Clique to Assignment)()()( zyxzyxzyxF

-x y z

-zz

y

x

-y

-x

G

y=0, z=1, x can take either values

NP-Completeness

Definition: A Language B is NP-complete if it satisfies two conditions:

1. B is in NP, and2. Every A NP is poly-time reducible to B.

Definition: A Language B is NP-hard if it satisfies the following condition:

2. Every A NP is poly-time reducible to B.

NP-completeness vs NP-hardness

P

NP

NP-complete

NP-hard

Two Theorems on NP-Completeness

Theorem 1: If B is NP-complete and BP then P=NP.

Theorem 2: If B is NP-complete and B P C for some C NP, then C is NP-complete.

Do NP-Complete Languages Exist?

Yes: BOUNDED HALTING (BH)

BH = {<M,x> | M is a TM and ∃c so that M(x,c) accepts in less than poly(|x|) steps}

Claim: BH is NP-complete.Proof: First, note that BH NP (why?).

Let A NP. We will show that A P BH.

A P BH

Since A NP, then it is verifiable in polynomial time. That is, there exists a poly-time TM V for which

w A iff ∃c s.t. V(w,c)= ACCEPT

Given w, the reduction f(w) generates the pair <V,w> as an instance for BH.

Need to show that <V,w> BH if and only if w A.

Correctness of Reduction

<V,w> BH

⇕ ∃c s.t. V(w,c) = ACCEPT

⇕ w A

Conclusion: There exists an NP-complete language.

Cook-Levin Theorem

Shows a “natural” NP-complete language.

Cook-Levin Theorem: SAT is NP-Complete.

Proof: We have to show:1. SAT NP2. For every A NP, A P SAT

The proof will be given next lecture.

Let’s See if we Understand…

Cook-Levin Theorem: SAT is NP-Complete.

Previous version: SAT ∈ P if and only if P = NP

We saw before: 3SAT PCLIQUE

You will see in tirgul: SAT P 3SAT

Question: What can we say about CLIQUE?

Showing that other languages are NP-complete

To show that B is NP-complete it is sufficient to show that:1. B is in NP, and2. Every A NP, APB

or2’. CPB, where C NP-complete.

Summary So Far

• P = problems that can be solved in poly-time• NP = problems for which a solution can be

verified in poly-time• Unknown whether P = NP (most conjecture

not)

• SAT is in NP:• Probably cannot be solved in polynomial time.• Easy to verify solution in polynomial time.

NP-Complete Problems are the “Hardest” Problems in NP

• If any one NP-Complete problem can be solved in polynomial time…

• …then every NP-Complete problem can be solved in polynomial time…

• …and in fact every problem in NP can be solved in polynomial time (which would prove that P = NP)

• Solve SAT in O(n100) time, you’ve proved that P = NP. Retire rich & famous.

Why Prove NP-completeness?

• Though nobody has proven that P ≠ NP, if you prove a problem is NP-Complete, most people accept that it is probably intractable.

• Therefore it can be important to prove that a problem is NP-Complete• Don’t try to come up with efficient algorithm.• Instead:

• Try to solve a subproblem • Work on approximation algorithms.

Using Reductions

Given one NP-Complete problem, can prove that many interesting problems are NP-Complete.

For example:• Graph problems (Clique, Vertex Cover, coloring) • Hamiltonian path/cycle• Knapsack problem• Traveling salesman• Job scheduling • Many, many, many more…

A vertex cover for a graph G is a set of vertices incident to every edge in G

VERTEX-COVER={<G,k>| G is an undirected graph that has a k-node vertex cover}

Theorem: vertex cover is NP-Complete

Example: The Vertex Cover Problem

Vertex Cover (Example)

A vertex cover of size 5 A vertex cover of size 4

Proof: The following V is a polynomial verifier for VERTEX-COVER.

V=“On input <<G,k>,c>:1. Test whether c is a collection of nodes in

G. 2. Test whether c has k different nodes.3. Test whether all edges in G touches the

nodes in c.4. If 1-3 pass, accept; Otherwise, reject.

VERTEX-COVER is in NP

Clique P Vertex Cover

Reduce k-clique to vertex cover:• The complement GC of a graph G contains

exactly those edges not in G

• Compute GC in polynomial time

• Claim: G has a clique of size k iff GC has a

vertex cover of size |V| - k

G Gc

Clique P Vertex Cover (cont’)

Claim 1: If G has a clique of size k, then GC has a vertex cover of size |V| - k

Claim 2: If GC has a vertex cover H V, with |H| = |V| - k, then G has a clique of size k

Proofs: In class

G Gc