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Unformatted text preview: 1 CS151 Complexity Theory Lecture 2 March 31, 2011 March 31, 2011 2 Reductions reductions are the main tool for relating problems to each other given two languages L 1 and L 2 we say L 1 reduces to L 2 and we write L 1 L 2 to mean: there exists an efficient (for now, polytime) algorithm that computes a function f s.t. x L 1 implies f(x) L 2 x L 1 implies f(x) L 2 March 31, 2011 3 Reductions positive use: given new problem L 1 reduce it to L 2 that we know to be in P . Conclude L 1 in P (how?) e.g. bipartite matching max flow formalizes L 1 as easy as L 2 yes no yes no L 1 L 2 f f March 31, 2011 4 Reductions negative use : given new problem L 2 reduce L 1 (that we believe not to be in P ) to it. Conclude L 2 not in P if L 1 not in P (how?) e.g. satisfiability graph 3coloring formalizes L 2 as hard as L 1 yes no yes no L 1 L 2 f f March 31, 2011 5 Reductions Example reduction: 3SAT = { : is a 3CNF Boolean formula that has a satisfying assignment } (3CNF = AND of OR of 3 literals) IS = { (G, k)  G is a graph with an independent set V V of size k } (ind. set = set of vertices no two of which are connected by an edge) March 31, 2011 6 Ind. Set is NPcomplete The reduction f: given = (x y z) ( x w z) () we produce graph G : x y z x w z one triangle for each of m clauses edge between every pair of contradictory literals set k = m 2 March 31, 2011 7 Reductions = (x y z) ( x w z) () Claim: has a satisfying assignment if and only if G has an independent set of size at least k proof? Conclude that 3SAT IS. x y z x w z March 31, 2011 8 Completeness complexity class C language L is Ccomplete if L is in C every language in C reduces to L very important concept formalizes L is hardest problem in complexity class C March 31, 2011 9 Completeness Completeness allows us to reason about the entire class by thinking about a single concrete problem related concept: language L is Chard if every language in C reduces to L March 31, 2011 10 Completeness May ask: how to show every language in C reduces to L? in practice, shown by reducing known C complete problem to L often not hard to find 1 st Ccomplete language, but it might not be natural March 31, 2011 11 Completeness Example: NP = the set of languages L where L = { x : 9 y, y x k , (x, y) R } and R is a language in P. one NPcomplete language bounded halting: BH = { (M, x, 1 m ) : 9 y, y x k s.t. M accepts (x, y) in at most m steps } challenge is to find natural complete problem Cook 71 : SAT NPcomplete March 31, 2011 12 Summary problems function, decision language = set of strings complexity class = set of languages efficient computation identified with efficient computation on Turing Machine singletape, multitape...
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 Fall '09

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