Towards the Resolution of Coppersmith-Sorkin Conjectures
نویسنده
چکیده
A k-matching is a set of k elements of a matrix, no two of which belong to the same row or column. The minimum weight k-matching of an m × n matrix C is the k-matching whose entries have the smallest sum. Coppersmith and Sorkin conjectured that if C is generated by choosing each entry independently from the exponential distribution of rate 1, then the expected value of the weight of the minimum weight k-matching is given by an explicit formula, whose proof is largely unknown. In this paper we describe our efforts to prove the Coppersmith-Sorkin conjecture by identifying the terms in the explicit formula to be the mean values of certain random variables which are functions of the matrix elements. We further conjecture that the distributions of these random variables are pure exponentials. We have partial theoretical backing and some simulation evidence for these conjectures. In the process we also prove a general combinatorial lemma about matchings in matrices.
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