Lecture 24 NP-Complete Problems

(1) Polynomial-time many-one reduction

A < m B p A set A in Σ* is said to be polynomial-time many-one reducible to B in Γ* if there exists a polynomial-time computable function f: Σ* → Γ* such that x ε A iff f(x) ε B.   Σ* Γ*          

A = Hamiltonian cycle (HC) Given a graph G, does G contain a Hamiltonian cycle?

B = decision version of Traveling Salesman Problem (TSP) Given n cities and a distance table between these n cities, find a tour (starting from a city and come back to start point passing through each city exactly once) with minimum total length. Given n cities, a distance table and k > 0, does there exist a tour with total length < k?

HC < m TSP p From a given graph G, we need to construct (n cities, a distance table, k).

3-SAT < m SAT p SAT: Given a Boolean formula F, does F have a satisfied assignment? An assignment is satisfied if it makes F =1. 3-SAT: Given a 3-CNF F, does F have a satisfied assignment?

Boolean Algebra

Boolean Algebra

3CNF

Examples

SAT < m 3-SAT p SAT: Given a Boolean formula F, does F have a satisfied assignment? An assignment is satisfied if it makes F =1. 3-SAT: Given a 3-CNF F, does F have a satisfied assignment?

Property of < m A < m B and B < m C imply A < m C A < m B and B ε P imply A ε P p p p

NP-complete A set A is NP-hard if for any B in NP, B < m A. A set A is NP-complete if it is in NP and NP-hard. A decision problem is NP-complete if its corresponding language is NP-complete. An optimization problem is NP-hard if its decision version is NP-hard. p

(2) Cook Theorem SAT is NP-complete

Proof of Cook Theorem

The 1st tape should be

The last tape should contain The final state.

Exercise!!!

3-SAT is NP-complete