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final(7) - CS 573 Final Exam Questions Fall 2010 This exam...

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CS 573 Final Exam Questions Fall 2010 This exam lasts 180 minutes. Write your answers in the separate answer booklet. Please return this question sheet with your answers. 1. A subset S of vertices in an undirected graph G is called triangle-free if, for every triple of vertices u , v , w S , at least one of the three edges uv , uw , vw is absent from G . Prove that finding the size of the largest triangle-free subset of vertices in a given undirected graph is NP-hard. A triangle-free subset of 7 vertices. This is not the largest triangle-free subset in this graph. 2. An n × n grid is an undirected graph with n 2 vertices organized into n rows and n columns. We denote the vertex in the i th row and the j th column by ( i , j ) . Every vertex in the grid have exactly four neighbors, except for the boundary vertices, which are the vertices ( i , j ) such that i = 1, i = n , j = 1, or j = n . Let ( x 1 , y 1 ) , ( x 2 , y 2 ) ,..., ( x m , y m ) be distinct vertices, called terminals , in the n × n grid. The escape problem is to determine whether there are m vertex-disjoint paths in the grid that connect the terminals to any m distinct boundary vertices. Describe and analyze an efficient algorithm to solve the escape problem. A positive instance of the escape problem, and its solution. 3. Consider the following problem, called U NIQUE S ET C OVER . The input is an n -element set S , together with a collection of m subsets S 1 , S 2 ,..., S m S , such that each element of S lies in exactly k subsets S i . Our goal is to select some of the subsets so as to maximize the number of elements of S that lie in exactly one selected subset. (a) Fix a real number p between 0 and 1, and consider the following algorithm: For each index i , select subset S i independently with probability p .
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