NP Completeness Piyush Kumar

Today Reductions Proving Lower Bounds revisited Decision and Optimization Problems SAT and 3-SAT P Vs NP Dealing with NP-Complete Problems –Average Case / Heuristics / Approx. Alg.

Convex Hulls and Sorting What are convex hulls? Given a program for computing convex hulls, can we use it to sort ?

Reduction from Sorting to CH –Given n real values x i, generate n 2D points on a parabola, e.g. (x i,x i 2 ). –Compute the (ordered) convex hull of the points. –The order of the convex hull points is the numerical order of the x i.

Reduction from Sorting to CH Does this imply a lower bound on computation of convex hulls?  (nlgn) ? Sorting  p CH  p : Can also be read as –“No harder to solve than”

Polynomial time reductions If P is polynomial-time reducible to Q, we will use the notation P  p Q P  p Q and Q  p R –Implies P  p R

Review : P, NP Set of problems : P, NP P = Can be solved in polynomial time NP = You can verify the solution in polynomial time P  NP ? P = NP ? (Probably not?)

definitions NP-Hard : For all R in NP, R  p T –Implies T is NP-Hard NP-Complete –T is both in NP and NP-Hard

Definitions : Revisited NP-Hard –If you can solve it, you can solve everything in NP NP-Complete –NP-Hard and in NP NP : Non-Deterministic Polynomial time. ( Guess and verify )

P NP NP- complete NP-hard Primality testing Semidefinite programming … Hamilton cycle, Independent Set, Vertex Cover, Graph 3-coloring, Satisfiability … Factoring … Halting Problem Assuming P  NP

Decision Vs Optimization Decision Problem: computational problem with intended output of “yes” or “no”, 1 or 0 Optimization Problem: computational problem where we try to maximize or minimize some value Introduce parameter k and ask if the optimal value for the problem is a most or at least k. Turn optimization into decision

Decision Vs Optimization Optimization Problem: TSP Decision problem –Is their a tour of length < k? Simplification: Worry only about decision problems. The class P, NP, NPC, NPH all constitute of only decision problems.

Cook’s Theorem Theorem (Cook 1970) SAT is NP-Complete. –CNF-SAT is also NP-Complete. Corollary 3-SAT is NP-Complete. Proof of Corollary –How do we prove it? –Recall the defn of NP-Complete

Proving NP Completeness What steps do we have to take to prove a problem P is NP-Complete? –A problem P is NP-complete if 1. it is in NP 2.  B  NP, B  p P (it is NP-Hard) –Idea for condition 2 : Use a known NP-Complete problem Q. –Show Q  p P. By transitivity of  p, we are done  B  NP, B  p Q  p P Hence :  B  NP, B  p P

Algorithm for P Reduction R R(x) Input for P Output for P yes/no Output for Q x Input for Q Reduction revisited: ( Q  P) Algorithm for solving Q

3-SAT is NP-Complete 3-SAT is in NP (Trivial) We will pick Q to be CNF-SAT To show : CNF-SAT  P 3-SAT. Let ∏ be the given CNF formula. We want to transform ∏ to an instance of 3-CNF. Idea: Transform each clause of ∏ so that it is 3-CNF

CNF-SAT to 3-SAT If less than 3 literals, repeat them. –Replace (a 1  a 2 ) by (a 1 ∨ a 2 ∨ a 2 ) If 3 literals, leave it as it is.

CNF-SAT to 3-SAT 4 literals : Replace (a 1  a 2  a 3  a 4 ) by (a 1  a 2  z)  (  z  a 3  a 4 ) In general, replace (a 1  a 2    a k ) by (a 1  a 2  z 1 )  (  z 1  a 3  z 2 )  (  z 2  a 4  z 3 )    (  z k-3  a k-1  a k ) – z 1, , z k-3 : new variables –This is analogous to the 4 literal case, just applied multiple times with more variables.

3-SAT is NP-Complete 3-SAT is in NP CNF-SAT  P 3-SAT –Since we can transform an input to the CNF-SAT problem in polynomial time to the 3-SAT problem solver and get a solution to CNF-SAT. Hence CNF- SAT is ‘no harder to solve than’ 3- SAT.

Other NP-Complete Problems Hamiltonian Cycle Of size k? –Vertex cover : A vertex cover of G is a set C  V such that all edges of E have at least one endpoint in C. –Clique : Find in a given graph G, if there is a size k clique?

Other NP-Complete Problems Set Cover : Given n sets, are there k sets out of these whose union is the same as the union of n sets? Subset Sum : Given a set of integers, and another integer k; Is there a subset of them that sum upto k?

Toy Problem TSP is NP-Complete –Decision Version: Does  a TSP with cost < k ? – Is TSP in NP? – What problem should we pick to prove this?

Approximation Algorithms For some NP-Complete problems, there are polynomial time algorithms that give near optimal solutions.

Vertex cover

VC is NP-Complete –(can be reduced from 3-SAT)

Approximation Alg for VC 1.S   2.E  E[G] 3.while E  4.let (u,v) be am arbitrary edge in E 5.S  S  {u,v} 6.remove from E every edge incident on either u or v 7.return S

example

2-Factor : Proof The vertex cover output by the approximation algorithm is no more than twice the size of the optimal vertex cover. Why?