lecture21 - CS 473 Algorithms Chandra Chekuri...

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Unformatted text preview: CS 473: Algorithms Chandra Chekuri [email protected] 3228 Siebel Center University of Illinois, Urbana-Champaign 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 NP-Complete 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 NP-Complete iff L is in NP for every L in NP, L ≤ P L L is NP-Hard if for every L in NP, L ≤ P L . Chekuri CS473ug Recap NP: languages that have polynomial time certifiers/verifiers A language L is NP-Complete iff L is in NP for every L in NP, L ≤ P L L is NP-Hard if for every L in NP, L ≤ P L . Theorem (Cook-Levin) Circuit-SAT and SAT are NP-Complete. Chekuri CS473ug Recap contd Theorem (Cook-Levin) Circuit-SAT and SAT are NP-Complete. Establish NP-Completeness via reductions: SAT ≤ P 3-SAT and hence 3-SAT is NP-complete 3-SAT ≤ P Independent Set (which is in NP) and hence Independent Set is NP-complete Vertex Cover is NP-complete Clique is NP-complete Set Cover is NP-Complete Chekuri CS473ug Today Prove Hamiltonian Cycle Problem is NP-Complete 3-Coloring is NP-Complete 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 NP-complete Directed Hamiltonian Cycle is in NP Chekuri CS473ug Directed Hamiltonian Cycle is NP-complete Directed Hamiltonian Cycle is in NP Certificate: Sequence of vertices Chekuri CS473ug Directed Hamiltonian Cycle is NP-complete 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 NP-complete 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 3-SAT ≤ P Directed Hamiltonian Cycle Chekuri CS473ug Reduction Given 3-SAT 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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lecture21 - CS 473 Algorithms Chandra Chekuri...

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