sol5 - AMS/MBA 546(Fall 2010 Estie Arkin Network Flows...

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AMS/MBA 546 (Fall, 2010) Estie Arkin Network Flows: Solution sketch to homework set # 5 1). Show that the following problems are NP-complete. You may use reductions from any of the problems shown NP-complete by Karp (3SAT, 3 dimensional matching, partition, vertex cover, Hamilton cycle/path, or clique), or any other problem discussed in class. Remember to also show the problem is in NP. (a). Given G = ( N,E ), is there a spanning tree T of G such that the nodes of T have degree at most 2? The problem is in NP. A succinct certi±cate for “yes” instances is such a tree, which can be veri±ed in polynomial time. Check whether the “tree” has n - 1 edges (and n nodes), and is it connected, thus showing it is a tree. Then we check the degree of each node to see if it is 1 or 2. All this can be done in time O ( n ). We reduce Hamilton Path to this problem. Given an instance of Hamilton Path, namely a graph G for which we want to know if a Hamilton Path exists, we ask (for the same graph) whether a spanning tree exists with all node degrees at most 2. Note that such a spanning tree is exactly a Hamilton Path! (b). Given

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This note was uploaded on 01/31/2011 for the course AMS 546 taught by Professor Arkin,e during the Fall '08 term at SUNY Stony Brook.

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sol5 - AMS/MBA 546(Fall 2010 Estie Arkin Network Flows...

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