CMPSCI 611: Advanced Algorithms Micah Adler Summary: CMPSCI 611: Advanced Algorithms Micah Adler Problem Set 2 Out: October 7, 2003 Due: October 14, 2003 1. For a graph G = (V; E), an edge cover E(V ) is a subset E 0  E of the edges that has at least one incident edge for every vertex in V . We can also de ne an edge cover for V 0  V , a subset of the vertices, where in this case the edge cover E(V 0 ) = E 0  E has at least one incident edge for every vertex in V 0 . The edges in E(V 0 ) are allowed to use vertices in V V 0 . Finally, a matching cover of V 0 is an edge cover of V 0 that is also a matching. (a) Given a graph G = (V; E), let (V; I) be the subset system where V 0 2 I if and only if there is a matching cover of V 0 . Show that this is indeed a subset system, and that it is a matroid. (b) Given a graph G = (V; E) and a coloring of its vertices (i.e., an assignment of one positive integer to each of its vertices), let (V; J ) be the subset system where V 0 2 J if and only if V 0 has at most one vertex of each color. Show that (V; J ) is a matroid. (c) Given a graph and a coloring of its vertices, show that we can nd in polynomial time the largest subset of vertices V 0 such that there is a matching cover of V 0 and there is at most one vertex of each color in V 0 . You can use without proof the fact that checking whether or not there is a matching cover of a subset V 0 can be done in time O(jV j  jEj). 2. In this problem we consider some generalizations of the knapsack problem. (a) Describe how to change the knapsack algorithm described in lecture to deal with the case where Collections: Computer Technologies and Information Sciences