MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems
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Let V be a set of n vertices, M a set of m labels, and let š be an m Ć n matrix of independent Bernoulli random variables with success probability p. A random instance G(V,E,š^Tš) of the weighted random intersection graph model is constructed by drawing an edge with weight [š^Tš]_v,u between any two vertices u,v for which this weight is larger than 0. In this paper we study the average case analysis of Weighted Max Cut, assuming the input is a weighted random intersection graph, i.e. given G(V,E,š^Tš) we wish to find a partition of V into two sets so that the total weight of the edges having one endpoint in each set is maximized. We initially prove concentration of the weight of a maximum cut of G(V,E,š^Tš) around its expected value, and then show that, when the number of labels is much smaller than the number of vertices, a random partition of the vertices achieves asymptotically optimal cut weight with high probability (whp). Furthermore, in the case n=m and constant average degree, we show that whp, a majority type algorithm outputs a cut with weight that is larger than the weight of a random cut by a multiplicative constant strictly larger than 1. Then, we highlight a connection between the computational problem of finding a weighted maximum cut in G(V,E,š^Tš) and the problem of finding a 2-coloring with minimum discrepancy for a set system Ī£ with incidence matrix š. We exploit this connection by proposing a (weak) bipartization algorithm for the case m=n, p=Ī(1)/n that, when it terminates, its output can be used to find a 2-coloring with minimum discrepancy in Ī£. Finally, we prove that, whp this 2-coloring corresponds to a bipartition with maximum cut-weight in G(V,E,š^Tš).
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