Random multi-index matching problems
O. C. Martin, M. Mezard, O. Rivoire
TL;DR
This paper applies the cavity method to analyze the properties of the multi-index matching problem with random costs, revealing a frozen glassy phase for dimensions greater than two and validating findings with small sample enumeration.
Contribution
It introduces a statistical physics approach to analyze the multi-index matching problem, extending understanding beyond traditional pairwise matchings.
Findings
For d>2, a frozen glassy phase with zero entropy is observed.
Theoretical predictions are supported by enumeration of small samples.
The study advances understanding of complex combinatorial optimization problems.
Abstract
The multi-index matching problem (MIMP) generalizes the well known matching problem by going from pairs to d-uplets. We use the cavity method from statistical physics to analyze its properties when the costs of the d-uplets are random. At low temperatures we find for d>2 a frozen glassy phase with vanishing entropy. We also investigate some properties of small samples by enumerating the lowest cost matchings to compare with our theoretical predictions.
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