Digraphs+Theory,+Algorithms+and+Applications_Part4 - 1.11...

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Unformatted text preview: 1.11 Exercises 43 1.67. 4-kings in bipartite tournaments. A vertex v in a digraph D is a k- king , if for every u ∈ V ( D )- { v } there is a ( v,u )-path of length at most k. Prove that a vertex of maximum out-degree in a strong bipartite tournament is a 4-king. For all s,t ≥ 4 construct strong bipartite tournaments with partite sets of cardinality s and t which do not have 3-kings. (Gutin [356]) 1.68. (+) A special case of the maximum independent set problem . The maximum independent set problem is as follows. Given an undirected graph G , find an independent set of maximum cardinality in G . The purpose of this exercise is to show that a special case of the maximum independent set problem is equivalent to the 2-satisfiability problem and hence can be solved using any algorithm for 2-SAT. (a) Let G = ( V,E ) be a graph on 2 k vertices and suppose that G has a perfect matching (i.e. a collection e 1 ,...,e k of edges with no common end-vertex). Construct an instance F of 2-SAT which is satisfiable if and only if G has an independent set of k vertices. Hint: fix a perfect matching M of G and let each edge in M correspond to a variable and its negation. (b) Prove the converse, namely if F is any instance of 2-satisfiability, then there exists a graph G = ( V,E ) with a perfect matching such that G has an independent set of size | V ( G ) | / 2 if and only if F is satisfiable. (c) Prove that it is NP-complete to decide if a given graph has an indepen- dent set of size at least ‘ , even if G has a perfect matching. Hint: use a reduction from MAX-2-SAT. 1.69. Linear time algorithm for finding an acyclic ordering of an acyclic digraph . Verify that the algorithm given in the proof of Proposition 1.4.3 can be implemented as an O ( n + m ) algorithm using the adjacency list rep- resentation. 1.70. Show how to extend the algorithm MergeHamPathTour (see Subsection 1.9.1) so that it works for tournaments with an arbitrary number of vertices. 1.71. Based on the proof of Theorem 1.5.1, give a polynomial algorithm to find cycles of lengths 3 , 4 ,...,n through a given vertex in a strong tournament T . What is the complexity of your algorithm and how do you store information about T and the cycles you find? 1.72. (+) Fast algorithm for Euler trails. Demonstrate how to implement the algorithm in the proof of Theorem 1.6.3 as an O ( n + m ) algorithm. Hint: use adjacency lists along with a suitable data structure to store the trail constructed so far. 1.73. Suppose S , T , K are decision problems such that S ≤ P T and T ≤ P K . Prove that S ≤ P K . 1.74. The independent set problem is as follows: Given a graph G = ( V,E ) and natural number k , decide whether G has an independent set of size at least k . Show that the independent set problem belongs to the complexity class NP ....
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Digraphs+Theory,+Algorithms+and+Applications_Part4 - 1.11...

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