# hwk6 - Fall 2011 CMSC 451 Homework 6 Clyde Kruskal Due at...

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Fall 2011 CMSC 451: Homework 6 Clyde Kruskal Due at the start of class Thursday, December 8, 2011. We know a number of problems are NP -complete including: Circuit SAT, SAT, 3-SAT, Independent Set, Vertex Cover, Hamiltonian Cycle, Traveling Salesman, 3-Dimensional Matching, Graph Coloring, Subset Sum, Clique, and Subgraph Isomorphism. Problem 1. HAMILTONIAN PATH PROBLEM: given a directed graph, does it contain a simple path that goes through every vertex exactly once? HAMILTONIAN CYCLE PROBLEM: given a directed graph, does it contain a di- rected simple cycle that goes through each vertex exactly once? Assume that the HAMILTONIAN PATH PROBLEM is known to be NP -complete. Given this assumption, prove that the HAMILTONIAN CYCLE PROBLEM is NP - complete. (Make sure to show that the HAMILTONIAN CYCLE PROBLEM is in NP .) Problem 2. Consider the problem DENSE SUBGRAPH: Given G , does it contain a sub- graph H that has exactly K vertices and at least Y edges? Prove that this problem is NP -complete.

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