On linear programs with random costs
Mathematical Programming: Series A and B
On the expected optimal value of random assignment problems: experimental results and open questions
Computational Optimization and Applications
The ζ (2) limit in the random assignment problem
Random Structures & Algorithms
Proofs of the Parisi and Coppersmith-Sorkin random assignment conjectures
Random Structures & Algorithms
The blind passenger and the assignment problem
Combinatorics, Probability and Computing
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A generalization of the random assignment problem asks the expected cost of the minimum-cost matching of cardinality k in a complete bipartite graph Km,n, with independent random edge weights. With weights drawn from the exponential distribution with intensity 1, the answer has been conjectured to beΣi,j≥0, i+jk1/(m−i)(n−j).Here, we prove the conjecture for k ⩽ 4, k = m = 5, and k = m = n = 6, using a structured, automated proof technique that results in proofs with relatively few cases. The method yields not only the minimum assignment cost's expectation but the Laplace transform of its distribution as well. From the Laplace transform we compute the variance in these cases, and conjecture that, with k = m = n → ∞, the variance is 2/n + O(log n/n2). We also include some asymptotic properties of the expectation and variance when k is fixed.