hw3 - CME 305 Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) HW#3 – Due 03/02/10 1. Recall the deﬁnition of totally unimodularity: a matrix A is totally unimodular if and only if the determinant of every submatrix of A is in {- 1 , 0 , 1 } . (a) Show that if every column of A has at most one +1 and at most one - 1, and all other entries are zero, then A is totally unimodular. (b) Let G ( V,E ) be a directed graph, and assume there is no edge entering s , and no edge leaving t . Write down a linear program for computing the maximum s - t ﬂow in G and show that the constraint matrix of the LP is totally unimodular. 2. Recall the maximum cut problem. Given a graph G ( V,E ) we want to partition the graph into two (disjoint) sets A and B such that the number of edges between them (the edges with one end point in A and the other in B ) is maximized. Consider the following algorithm. We start with an arbitrary partition ( A,B ). If there exists a vertex
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