ps9 - Introduction to Algorithms Massachusetts Institute of...

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Unformatted text preview: Introduction to Algorithms Massachusetts Institute of Technology Professors Erik Demaine and Shafi Goldwasser December 6, 2002 6.046J/18.410J Handout 31 Problem Set 9 (Optional) This problem set is not due; it is optional. Reading: Chapter 34 Problem 9-1. Prove that the following problems are in Problem 9-2. A subgraph of a graph is a graph where ; i.e. it is a subset of the vertices together with all the edges of the original graph which are incident to these vertices. Consider the problem L ARGEST C OMMON S UBGRAPH : Given two graphs and and an integer , determine whether there is a graph with edges which is a subgraph of both and . (Hint: Reduce from C LIQUE .) is a collection of edges Problem 9-3. A perfect matching in an undirected graph such that each node has exactly one edge of incident to it. (In other words, the degree of each node in is exactly one.) Given a weight on each edge , there is an algorithm to find a perfect matching in of minimum total weight; you may use such an algorithm as a subroutine in your solution to the following problem, without worrying about how it works. Give an algorithm for the traveling salesman problem with triangle inequality that produces a solution within a factor of 3/2 of the optimal one. (Hint: The key to the first algorithm given in class is to convert a minimum spanning tree to an Eulerian graph, and then shortcut that to obtain a tour; find a way to do this in a less expensive way by using matching.) Problem 9-4. Here we will see how a decision algorithm for an used to efficiently find a witness for that problem (and others). (a) Prove that if C LIQUE graph and an integer “NONE” otherwise. -complete problem can be , then there is a polynomial-time algorithm that takes a and finds a clique in of size if one exists, and outputs (b) Under the same assumption (C LIQUE ), prove that there is a polynomial-time algorithm that takes a set of cities and the distances between them and finds a minimum length traveling salesman tour through all the cities. Bu 5 Atsrq D c@ ¤ ¥ BD7 SD 5 I @D 5 ¦D WVBRUTR87QPAHG7F§E ) G  (b) it is possible to assign one of three “colors” to each vertex of no two neighboring vertices are assigned the same color B pihWgf5 ¤ `  B @ 5 ¦ c987b© ¡ v  ! Yx ' B5 pihWgf0e  ! ba  B @ 5 ¦ CA98764 ¡w D c@   s ! ¡w % ¨ ¦ ) 3&210£ ' (! %# !    ¨ ¦ ¤ &$" ©§¥£ @ 5 DB A§7d¦ D   (a) there exists a simple path in ¡ ¢ : and , of length at least such that between 7 XD Y(G7 ! ) R 2 Handout 31: Problem Set 9 (Optional) Problem 9-5. Consider the following definition of a randomized reduction: (probability taken over choice of (One way to interpret this definition is that represents some random choices used in the reduction. That is, rather than letting make random choices on its own, we give it a random string to use for this purpose.) CC 6 5  @ ED¤ 7!BA , where Now, suppose (polynomial time) algorithm for (i.e., has a poly-time algorithm). Describe a randomized that is correct with probability at least .     " ! 8'  ¨ w  ! ¥ £ £¡ 9¢x ¡w£ ¥  6 x2£ 75!431w B   § 5 ¥ 0'(¡  %&¨¦ ) © w§ £$w B   § 5 # " ! '  ¨ w ¤© w ¨¦ ¥ §  £¡ ¤¢x if there exists a polynomial time function and a constant such that ) ...
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This note was uploaded on 01/01/2011 for the course CS 5503 taught by Professor Charlese.leiserson during the Fall '01 term at MIT.

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