Homework 8 Solution


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1. Suppose 2n teams play in a round-robin tournament. Over a period of 2n 1 days, every team plays every other team exactly once. There are no ties. Show that for each day we can select a winning team, without selecting the same team twice. (Hint: Consider a bipartite graph with a set of team vertices and a set of day vertices. Take any set of days W and assume not all teams won in some day in W . Let tw be a team that did not win in any day in W . Consider the implication on the number of teams that won at least once in some day in W . Invoke Hall’s Theorem.)

  1. Given an undirected graph G = (V, E) and an integer k. A clique of G is a subset V 0 V of vertices, each pair of which is connected by an edge in E. The Clique problem asks whether G contains a clique of size at least k. An independent set of G is a subset V 0 V of vertices such that each edge in E is incident on at most one vertex in V 0. The Independent-Set problem asks whether G0 contains an independent set of size at least k0. We proved in class that the Clique problem is NP-complete. Show that the independent set problem is same by reduction from Clique problem.

  1. Show that the following three problems are polynomial time reducible to each other.

Set-Cover: Given a collection of sets, and a number k, the Set-Cover problem asks if there are at most k sets from the collection of sets such that their union contains every element in the union of all sets.

Hitting-Set: Given a collection of sets, and a number k, the Hitting-Set problem asks if are there at most k elements of the sets such that there is at least one element from each set?

Dominating-Set: Given an undirected graph G, and a number k, the Dominating-Set prob-lem asks if there is a subset of vertices of size k such that every vertex in the graph is either in the subset or has a neighbor that is in the subset.

Prove Set-Cover, Hitting-Set and Dominating-Set are polynomial-time reducible to each other.

(Hint: One strategy is to show Set-Cover p Hitting-Set, Hitting-Set p Dominating-Set and Dominating-Set p Set-Cover. An alternative way is to show Hitting-Set p Dominating-Set, Dominating-Set p Hitting-Set, Set-Cover p Dominating-Set and Dominating-Set p Set-Cover. In class we have seen Vertex-Cover reduced poly to Dominating-Set).

  1. Given a directed graph G = (V, E) and a pair of vertices s, t in G, the Hamiltonian Path problem asks whether there is a simple path from s to t that visits every vertex of G exactly once. The Hamiltonian Cycle problem asks if there is a cycle in a directed graph G that visits every vertex exactly once. Show that Hamiltonian Path and Hamiltonian Cycle problems are polynomial-time reducible to each other.

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