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Unformatted text preview: CS 473: Algorithms Chandra Chekuri [email protected] 3228 Siebel Center University of Illinois, UrbanaChampaign Fall 2009 Chekuri CS473ug Recap NP: languages that have polynomial time certifiers/verifiers Chekuri CS473ug Recap NP: languages that have polynomial time certifiers/verifiers A language L is NPComplete iff L is in NP for every L in NP, L ≤ P L Chekuri CS473ug Recap NP: languages that have polynomial time certifiers/verifiers A language L is NPComplete iff L is in NP for every L in NP, L ≤ P L L is NPHard if for every L in NP, L ≤ P L . Chekuri CS473ug Recap NP: languages that have polynomial time certifiers/verifiers A language L is NPComplete iff L is in NP for every L in NP, L ≤ P L L is NPHard if for every L in NP, L ≤ P L . Theorem (CookLevin) CircuitSAT and SAT are NPComplete. Chekuri CS473ug Recap contd Theorem (CookLevin) CircuitSAT and SAT are NPComplete. Establish NPCompleteness via reductions: SAT ≤ P 3SAT and hence 3SAT is NPcomplete 3SAT ≤ P Independent Set (which is in NP) and hence Independent Set is NPcomplete Vertex Cover is NPcomplete Clique is NPcomplete Set Cover is NPComplete Chekuri CS473ug Today Prove Hamiltonian Cycle Problem is NPComplete 3Coloring is NPComplete Chekuri CS473ug Directed Hamiltonian Cycle Input Given a directed graph G = ( V , E ) with n vertices Goal Does G have a Hamiltonian cycle ? Chekuri CS473ug Directed Hamiltonian Cycle Input Given a directed graph G = ( V , E ) with n vertices Goal Does G have a Hamiltonian cycle ? A Hamiltonian cycle is a cycle in the graph that visits every vertex in G exactly once Chekuri CS473ug Directed Hamiltonian Cycle Input Given a directed graph G = ( V , E ) with n vertices Goal Does G have a Hamiltonian cycle ? A Hamiltonian cycle is a cycle in the graph that visits every vertex in G exactly once Chekuri CS473ug Directed Hamiltonian Cycle is NPcomplete Directed Hamiltonian Cycle is in NP Chekuri CS473ug Directed Hamiltonian Cycle is NPcomplete Directed Hamiltonian Cycle is in NP Certificate: Sequence of vertices Chekuri CS473ug Directed Hamiltonian Cycle is NPcomplete Directed Hamiltonian Cycle is in NP Certificate: Sequence of vertices Certifier: Check if every vertex (except the first) appears exactly once, and that consecutive vertices are connected by a directed edge Chekuri CS473ug Directed Hamiltonian Cycle is NPcomplete Directed Hamiltonian Cycle is in NP Certificate: Sequence of vertices Certifier: Check if every vertex (except the first) appears exactly once, and that consecutive vertices are connected by a directed edge Hardness: We will show 3SAT ≤ P Directed Hamiltonian Cycle Chekuri CS473ug Reduction Given 3SAT formula ϕ create a graph G ϕ such that G ϕ has a Hamiltonian cycle if and only if ϕ is satisfiable G ϕ should be constructible from ϕ by a polynomial time algorithm A Notation: ϕ has n variables x 1 , x 2 ,..., x n and m clauses C 1 , C 2 ,..., C m ....
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 Fall '08
 Chekuri,C
 Algorithms

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