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Unformatted text preview: ECS 222A: Algorithm Design and Analysis Handout ?? UC Davis — Charles Martel December. 11, 2008 Final Exam Instructions: Please write clearly and succinctly; be sure to give an overview before plunging into details. Note that if an algorithm is requested, the most efficient one is desired. Your name: On problem you got out of 1 20 2 22 3 24 4 17 5 17 Σ 100 ECS 222A Handout ?? : Final Exam 2 1 Independent Set [20 points] The standard Independent Set (IS) problem takes as input an undirected graph G=(V,E) and finds a maximum size subset S of V such that no edge in E connects two vertices in S. We now consider a variant of of IS, min violation IS, MVIS , where we are still given given an undirected graph G=(V,E) but we now have non-negative edge weights ( w ( i, j ) for ( i, j ) in E , and w ( i, j ) = 0 if ( i, j ) not in E) and a target set size k . We want a set S of k vertices such that the sum of the edge weights connecting vertices in S is minimum (note: an IS S has weight zero since no edges connect vertices). a) Show that MVIS is NP-hard. b) Suppose our graph G is a binary tree (undirected, but with a root r ). Give a polynomial time algorithm to solve the MVIS problem using dynamic programming. You need only describe how to find the minimum cost of a solution set S, not the actual set S. You may find it useful to compute for each node v OPT( v, i ) the min-cost solution for the subtree rooted at v that includes i nodes, for i = 0 , 1 , . . . k . You will likely also want to compute other values for node v as well. Please define those you use carefully....
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This note was uploaded on 01/21/2010 for the course CS 210 taught by Professor Chip during the Fall '09 term at UC Davis.
- Fall '09