On the number of $k$-cycles in the assignment problem for random matrices

TL;DR

This paper investigates how the structure of random cost matrices influences the prevalence of k-cycles in optimal solutions of the assignment problem, revealing symmetry-dependent cycle distributions through analysis and modeling.

Contribution

It provides a detailed analysis of the number of k-cycles in the assignment problem for symmetric and antisymmetric random matrices, including an analytic model explaining the numerical findings.

Findings

Symmetric matrices favor one- and two-cycles in solutions.

Antisymmetric matrices suppress one- and two-cycles.

A simple ansatz accurately predicts cycle counts based on matrix symmetry.

Abstract

We continue the study of the assignment problem for a random cost matrix. We analyse the number of $k$-cycles for the solution and their dependence on the symmetry of the random matrix. We observe that for a symmetric matrix one and two-cycles are dominant in the optimal solution. In the antisymmetric case the situation is the opposite and the one and two-cycles are suppressed. We solve the model for a pure random matrix (without correlations between its entries) and give analytic arguments to explain the numerical results in the symmetric and antisymmetric case. We show that the results can be explained to great accuracy by a simple ansatz that connects the expected number of $k$-cycles to that of one and two cycles.