Lecture 23 NPHard Problems Outline NP NPhard NPcomplete

Lecture 23 NP-Hard Problems

Outline • NP, NP-hard, NP-complete • Cook-Levin Theorem, first NP-hard problems • Reductions • INDEPENDENT SET to CLIQUE • 3 -SAT to INDEPENDENT SET

NP-hard problems • A problem A is NP-hard, if for all problem B in NP, B can be reduced to A in polynomial time. • A is “harder” than all problems in NP, hence NP-hard. • A problem A is NP-complete, if it is in NP and also NP-hard. • NP-hard but not NP-complete? • May not be a decision problem, e. g. longest path. • May be even harder than NP-complete problems.

Relationships NP-hard NP-complete NP P

NP-complete problems • Claim: All NP-complete problems are equally hard. For any two NP-complete problems A, B, A can be reduced to B and B can be reduced to A. • Claim: If any NP-complete problem has a polynomial time algorithm, then P = NP. • Claim: If A can be reduced to B, B can be reduced to C, then A can be reduced to C. • How do we prove a problem is NP-complete?

Outline • NP, NP-hard, NP-complete • Cook-Levin Theorem, first NP-hard problems • Reductions • INDEPENDENT SET to CLIQUE • 3 -SAT to INDEPENDENT SET

The first NP-complete problem • Gates • Circuits

CIRCUIT-SAT problem • Given a circuit with n inputs and 1 output, is there a possible input (represented by n-bit binary string) that makes the output 1? • Clearly in NP: • Prover gives the input as an n-bit string, • Verifier follows the circuit and computes the output, check that it is indeed 1. • Theorem[Cook-Levin] CIRCUIT-SAT is NP-complete.

Proof idea of Cook-Levin Verify(color[]) //color[] is an array with 0, 1, 2 FOR each edge (u, v) IF color[u]<>color[v] THEN RETURN FALSE RETURN TRUE NP-hard problem Verifier

Proof idea of Cook-Levin Verify(color[]) //color[] is an array with 0, 1, 2 FOR each edge (u, v) IF color[u]<>color[v] THEN RETURN FALSE RETURN TRUE Compiler Input: binary Output: instance! encoding. CIRCUIT-SAT of True/False color[] 01010101 0010101011 Machine code

Outline • NP, NP-hard, NP-complete • Cook-Levin Theorem, first NP-hard problems • Reductions • INDEPENDENT SET to CLIQUE • 3 -SAT to INDEPENDENT SET

Other NP-hard Problems A NP-Hard • • Reduction B NP-Hard A is NP-hard For any problem C in NP, C is easier than A. A can be reduced to B A is easier than B For any problem C in NP, C is easier than B B is NP-hard

General Recipe for reductions • In order to prove B is NP-hard, given that we know A is NP hard. • Reduce A to B. • What is a complete reduction? 1. Given an instance X of A, construct an instance Y of B. 2. Prove that if answer to X is YES, answer to Y is also YES. 3. Prove that if answer to X is NO, answer to Y is also NO. (Usually: prove the contrapositive: if answer to Y is YES, answer to X is also YES)

INDEPENDENT SET • Given a graph, decide whethere is a set of k vertices, such that no two vertices in the set are connected by an edge.

CLIQUE • Given a graph, decide whethere is a set of k vertices, such that all pairs of vertices in the set are connected by an edge. • Claim: INDEPENDENT SET can be reduced to CLIQUE.

Reduction from INDEPENDENT SET to CLIQUE • Step 1: Given an instance X of INDEPENDENT SET, construct an instance Y of CLIQUE. • Idea: Find similarities/differences of the problems. • INDEP. SET: find vertices such that they are not connected. • CLIQUE: find vertices such that they are connected.

3 -SAT •

Reduction from 3 -SAT to INDEPENDENT SET • Step 1: Given an instance X of 3 -SAT, construct an instance Y of INDEPENDENT SET. • Problem: the two problems look totally different. • Idea: Construct “Gadgets”: for each concept in 3 SAT (variables, clauses), construct a part of instance of INDEP. SET (some vertices/edges) so that we can connect the two problems.

Gadget for variables • Variable xi: either true or false • Gadget: A graph with two different independent sets, corresponding to xi = true or false. • Many constructions, but easiest one ui vi • Hope: ui in independent set xi = true vi in independent set xi = false

Gadget for Clauses •

Gadget for Clauses Goal: When clause is not satisfied, cannot select any vertex in gadget. v 1 v 3 u 5 Gadget When clause is satisfied, can select exactly one vertex in gadget This encourages all the clauses to be satisfied.

Gadget for Clauses v 1 v 3 u 5 w 1 w 2 w 3 Goal: When clause is not satisfied, cannot select any vertex in gadget. When clause is satisfied, can select exactly one vertex in gadget This encourages all the clauses to be satisfied.

Reductions • Claim: CIRCUIT-SAT can be reduced to 3 -SAT. (therefore 3 -SAT is also NP-complete). • Claim: 3 -SAT can be reduced to INDEPENDENT SET. CIRCUIT -SAT NP-complete 3 -SAT NP-complete INDSET NP-complete CLIQUE NP-complete