The Random Fractional Matching Problem

TL;DR

This paper analyzes two variants of the random-link fractional matching problem, deriving finite-size corrections to the asymptotic optimal cost using replica methods and comparing results with simulations.

Contribution

It provides the first analytical derivation of finite-size corrections for the fractional matching problem using replica techniques.

Findings

Both formulations share the same asymptotic average optimal cost.

Finite-size corrections are analytically derived and validated against simulations.

Differences between fractional and integer matching problems are discussed.

Abstract

We consider two formulations of the random-link fractional matching problem, a relaxed version of the more standard random-link (integer) matching problem. In one formulation, we allow each node to be linked to itself in the optimal matching configuration. In the other one, on the contrary, such a link is forbidden. Both problems have the same asymptotic average optimal cost of the random-link matching problem on the complete graph. Using a replica approach and previous results of W\"{a}stlund [Acta Mathematica 204, 91-150 (2010)], we analytically derive the finite-size corrections to the asymptotic optimal cost. We compare our results with numerical simulations and we discuss the main differences between random-link fractional matching problems and the random-link matching problem.