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Moment generating functions in combinatorial optimization: Bipartite matching

Johan Wästlund

TL;DR

The work develops a recursive framework to characterize the full distribution of minimum-cost matchings on complete bipartite graphs with exponential edge costs via ghost-vertex constructions and ghost-generating functions. It proves the recursion for these MGFs, illustrates with a 3-by-3 example, and shows how moments and cumulants follow from series expansions, linking the analytic structure to potential Gaussian scaling in the large-$n$ limit. The authors conjecture a zero-free disk for the MGFs and provide partial results supporting a Gaussian limit for the standard assignment problem, along with asymptotics for fixed $k$ and large $m,n$. This framework yields a computational pathway for finite-n distributions and sheds light on the asymptotic behavior and zeros of the moment generating functions in random bipartite matching.

Abstract

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.

Moment generating functions in combinatorial optimization: Bipartite matching

TL;DR

The work develops a recursive framework to characterize the full distribution of minimum-cost matchings on complete bipartite graphs with exponential edge costs via ghost-vertex constructions and ghost-generating functions. It proves the recursion for these MGFs, illustrates with a 3-by-3 example, and shows how moments and cumulants follow from series expansions, linking the analytic structure to potential Gaussian scaling in the large- limit. The authors conjecture a zero-free disk for the MGFs and provide partial results supporting a Gaussian limit for the standard assignment problem, along with asymptotics for fixed and large . This framework yields a computational pathway for finite-n distributions and sheds light on the asymptotic behavior and zeros of the moment generating functions in random bipartite matching.

Abstract

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.
Paper Structure (21 sections, 13 theorems, 105 equations, 6 figures)

This paper contains 21 sections, 13 theorems, 105 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Minimum cost matching
  3. Finite $n$ results for exponential edge costs
  4. Conjectured Gaussian limit
  5. Cumulants
  6. The recursion and its applications
  7. Matchings of $k$ edges on $K_{m,n}$
  8. Adding a ghost vertex
  9. Recursion
  10. Example: 3 by 3 perfect matching
  11. Computing the first few moments of $C_{k,m,n}$
  12. Proof of the recursion
  13. Integrating over the cost of a ghost edge
  14. Nesting and locality
  15. Independence lemmas
  16. ...and 6 more sections

Key Result

Lemma 1

Figures (6)

  • Figure 1: The $6=3!$ perfect matchings on the complete bipartite graph $K_{3,3}$. The $9=3^2$ edges are assigned independent mean 1 exponential costs, and the minimum total cost $C_3$ of a perfect matching has expectation $1+1/4+1/9=49/36$ according to equation \ref{['parisi']}. We show in the following how to characterize the distribution of $C_n$ completely.
  • Figure 2: The density functions $f_n$ of $C_n$ for $n=1,\dots, 25$. We recognize $f_1$ as the one with $f_1(0)=1$, and $f_2$ by $f'_2(0)>0$. Then $f_3,\dots, f_{25}$ follow in order of increasing density around $\pi^2/6\approx 1.64$.
  • Figure 3: The density function $(9x^4/4 + 12x^3 - 15x^2 + 36x -36) e^{-3x} + 36e^{-4x}$ of the cost $C_{3,3,3}$ of the 3 by 3 minimum perfect matching.
  • Figure 4: The graph of $C_{k,m,n}$ as a function of $x$.
  • Figure 5: The $190 = \binom{20}{2}$ complex zeros of the moment generating function $F_{20,20,20}$ of the minimum cost of a 20 by 20 perfect matching. There are 28 real zeros of which the smallest is at approximately $20.04$ and the largest at $137.80$, and 81 complex conjugate pairs. The first pole of $F_{20,20,20}$ is at $t=20$ and has order 39. There are several zeros close to this pole but none in the disk of radius 20 around the origin. Even so, there is a pair of "overhang" zeros whose real part is negative, and in particular well below 20.
  • ...and 1 more figures

Theorems & Definitions (25)

  • Lemma 1
  • proof : Proof of Lemma \ref{['L:integration']}
  • Lemma 2: Locality
  • proof
  • Lemma 3: Nesting
  • proof
  • Lemma 4: Ghost locality
  • Lemma 5: Ghost nesting
  • Lemma 6
  • proof
  • ...and 15 more