# ps8 - 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 May 7, 2004 6.046J/18.410J Handout 24 Problem Set 8 (Optional) This problem set is not due and is meant as practice for the ﬁnal. Reading: 26.1, 26.3, 35.1 Problem 8-1. Prove these problems are NP-Complete: R P  D ¢ QIHGFECB ¡'%&\$¡ # ¡ !¦¦©§¥£¡   ¨ ¦¤ ¢ (a) S ET-C OVER : Given a ﬁnite set , a collection , and an integer , determine whether there is a sub-collection of that covers . In other words, determine whether there exists and . of subsets of with cardinality such that FV  XWUTSR ¢ 2 A@¦8976543\$1 " "0¢ # ( )¡ ( " and two dis(b) D IRECTED -H AMILTONIAN -PATH : given a directed graph tinct vertices , determine whether contains a path that starts at , ends at , and visits every vertex of the graph exactly once. (Hint: Reduce from H AM - CYCLE : 34.5.3 in CLRS.) T B Problem 8-2. M AX -C UT Approximation P  ¢ QIHGFD ¥B ¦ a tsF I VP r&qUTSpRD P  D ¢ gIHGFfeB ¦ a ihF P¦ a  dcb`FYG¦D A cut of an undirected graph is a partition of V into two disjoint subsets and . We say that an edge crosses the cut if one of its endpoints is in and the other is in . The MAX - CUT problem is the problem of ﬁnding a cut of an undirected connected graph that maximizes the number of edges crossing the cut. Give a deterministic approximation algorithm for this problem with a ratio bound of . Hint: Your algorithm should guarantee that the number of edges crossing the cut is at least half of the total number of edges. P¦ a  dch`FYG¦D u ¦ ¦ Problem 8-3. Global Edge Connectivity of Undirected and Directed Graphs (a) The global edge connectivity of an undirected graph is the minimum number of edges that must be removed to disconnect the graph. Show how the edge connectivity of an undirected graph can be determined by running the maximum-ﬂow algorithm times, each on a ﬂow network with vertices and edges. P ( I (  D P ( F (  D P  D ¢ gIHGFwvB B x ya ( F ( (b) The global edge connectivity of a directed graph is the minimum number of directed edges that must be removed from so that the resulting graph is no longer strongly connected. Show how the edge connectivity of a directed graph can times, each on a ﬂow be determined by running the maximum-ﬂow algorithm vertices and edges. network with P  gIHGFD ¢ B ( F( B P ( I (  D P ( F (  D Problem 8-4. Perfect Matching in Regular Bipartite Graph  F  ¢ , where , is -regular if every vertex F V T P  D ¢ gIHGFB A bipartite graph exactly . has degree  Handout 24: Problem Set 8 (Optional) 2 ( ( ¢ ( (  (a) Prove that for every -regular bipartite graph, . (b) Model the maximum d-regular bipartite matching as a max-ﬂow problem as in Section 26.3 in CLRS. Show that the max-ﬂow value from to in the formulation is . ( ( ( ( ¡  (c) Prove that every -regular bipartite graph has a matching of cardinality . ...
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## This note was uploaded on 10/08/2009 for the course EECS 6.046 taught by Professor Erikdemaine during the Spring '04 term at MIT.

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