CSE 6311 Spring 2009 ADVANCED COMPUTATIONAL MODELS AND

CSE 6311  Spring 2009 ADVANCED COMPUTATIONAL MODELS AND

CSE 6311 Spring 2009 ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS Lecture Notes Feb. 3, 2009 Instructor: Dr. Gautam Das notes by Walter Wilson Some NP-Complete problems: SAT, 3SAT, CLIQUE, VERTEX COVER, MAX INDEP SET SAT - Satisfiability:

Inputs: a) n Boolean variables: v1,..,vn b) Boolean formula variables & operators (and &, or |, not ~) Example: (v1 | ~v2) & (~v1 | v2 | v3) Output: Is there a satisfying assignment to the variables? For our example yes: v1=1, v2=0, v3=1 Cook's Theorem (or Cook-Levin Thm) "The complexity of theorem proving

procedures" (1971) SAT is NP-complete: var asmt can be verified in polynomial time Boolean expr satisfied iff nondeterministic Turing machine accepts NP problem Karp - reduction "Reducibility Among Combinatorial Problems" (1972) Proved 21 NP-complete problems 3SAT Inputs:

a) n Boolean variables: v1,..,vn b) 3-CNF (Conjunctive Normal Form) Boolean formula: Conjunction of clauses, each clause a disjunction of 3 literals, each literal a variable or its negation (if clause length = 2, problem is P) (3 is smallest length that is NP) CLIQUE largest complete subgraph Decision problem: Given graph G (n vertices, m edges) and k>0, is there a clique of size >=k? Naive algorithm look at all 2^n vertex subsets

Problem in NP verification of clique is in P Reduce NP-complete problem 3SAT to CLIQUE: Create vertex for each literal of each clause Make edges between vertices of different clauses except for opposite literals Clique of size C exists iff 3SAT formula is satisfiable where C is number of clauses Thus CLIQUE is NP-complete VERTEX COVER Smallest vertex subset that touches all edges

Decision problem: Given graph G (n vertices, m edges) and k>0, is there a cover of size <=k? Reduce CLIQUE to VERTEX COVER: Given Clique inputs G,k compute VC inputs: G' = complement of G (edge (vi,vj) in G' iff not in G) k' = n - k Clique of size k in G iff VC of size k' in G' Example graph: a

c b d g e f Vertex Cover: {a, g, e} MAX INDEP SET Independent Set: vertex subset with no edges within the set

Decision problem: Given graph G and k, is there an independent set of size >=k? Is in NP: easy to verify subset as independent Reduce VERTEX COVER to MAX INDEP SET Given VC inputs G,k compute Max Indep. Set inputs: G' = G k' = n - k VC of size k iff Independent Set of size k' Max Indep Set = {b, c, d, f} for example graph

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