1. CSE 6311 – Spring 2009 ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS Lecture 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. Example graph: a b c d e f g 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