3/24/08 - NP-complete sequencing pro. .. NP: problems whose solution can be e f ciently (poly-time) verifed. Ex: Compositeness: Given n-bit number x, in x composite? “Hint”: a pair y,z > 1 such that yz = x. NP-Complete: a problem X which is in NP and every other problem Y in NP has a poly-time reduction FROM Y TO X. Ex. Ind. Set, 3 SAT. Alt de±: a problem X which is in NP and there exists NPC problem Y which has a poly-time reduction ±rom Y to X. 2 easy reductions ±rom ind. set. (1) CLIQUE: Given undirected graph G = (V, E) and a number k > 0. Output “yes” i± G has a subgraph H with k vertices and every 2 vertices o± H are joined by an edge. (H is a “k-clique”.) Take (G, k) an instance o± IND. SET. Construct G ̅ which has same vertex set and (u,v) ∈ E(G ̅ ) <=> (u,v) ∉ E(G). (1) Reduction runs in poly-time. YES! O(n 2 ) (2) I± G has k-ind set, G ̅ has a k-clique. (3) I± G has no k-ind set, G
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