Moment generating functions in combinatorial optimization: Bipartite matching

In a random model of minimum cost bipartite matching based on exponentially distributed edge costs, we show that the distribution of the cost of the optimal solution can be computed efficiently. The distribution is represented by its moment generating function, which in this model is always a rational function. The complex zeros of this function are of interest as the lack of zeros near the origin indicates a certain regularity of the distribution. We propose a conjecture according to which these moment generating functions never have complex zeros of smaller modulus than their first pole. For minimum cost perfect matching, also known as the assignment problem, such a zero-free disk would imply a Gaussian scaling limit.

💡 Research Summary

In this paper the author studies the classic assignment problem on a complete bipartite graph Kₙ,ₙ where each of the n² edges receives an independent exponential cost with mean 1. The main object of interest is the random total cost Cₙ of the minimum‑cost perfect matching. While earlier work had identified the limiting mean (π²/6) and variance (4ζ(2)−4ζ(3)) of Cₙ, this work goes far beyond by giving a complete description of the entire distribution for every finite n.

The key technical device is the moment‑generating function (MGF) Fₙ(t)=E