CSE 6311 – Spring 2009 ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS Lecture Notes – Feb. 3, 2009

1. CSE 6311 – Spring 2009ADVANCED COMPUTATIONAL MODELS AND ALGORITHMSLecture Notes – Feb. 3, 2009 Instructor: Dr. Gautam Das notes by Walter Wilson

2. 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

3. 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

4. 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)

5. 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

6. 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'

7. a c b Example graph: d g e f Vertex Cover: {a, g, e}

8. 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