A frozen glass phase in the multi-index matching problem

TL;DR

This paper investigates the thermodynamics of the multi-index matching problem, revealing a complex phase diagram with a finite temperature transition to a frozen glass phase, using the cavity method and numerical validation.

Contribution

It introduces a detailed thermodynamic analysis of the multi-index matching problem, uncovering a glass phase transition not present in the bipartite case.

Findings

Identification of a finite temperature phase transition to a frozen glass phase

Derivation of critical temperature and ground state energy density

Validation of theoretical results with numerical studies

Abstract

The multi-index matching is an NP-hard combinatorial optimization problem; for two indices it reduces to the well understood bipartite matching problem that belongs to the polynomial complexity class. We use the cavity method to solve the thermodynamics of the multi-index system with random costs. The phase diagram is much richer than for the case of the bipartite matching problem: it shows a finite temperature phase transition to a completely frozen glass phase, similar to what happens in the random energy model. We derive the critical temperature, the ground state energy density, and properties of the energy landscape, and compare the results to numerical studies based on exact analysis of small systems.