Scaling and Universality in Continuous Length Combinatorial Optimization

TL;DR

This paper investigates how solution costs in various combinatorial optimization problems change under small perturbations, revealing different scaling behaviors that suggest a universality classification akin to statistical physics.

Contribution

It identifies distinct scaling laws for solution cost increases in different optimization problems and proposes a universality framework for their classification.

Findings

Minimum spanning tree cost scales as delta^2.

Matching and TSP problems scale as delta^3 in higher dimensions.

Monte Carlo simulations and theoretical analysis support the scaling laws.

Abstract

We consider combinatorial optimization problems defined over random ensembles, and study how solution cost increases when the optimal solution undergoes a small perturbation delta. For the minimum spanning tree, the increase in cost scales as delta^2; for the mean-field and Euclidean minimum matching and traveling salesman problems in dimension d>=2, the increase scales as delta^3; this is observed in Monte Carlo simulations in d=2,3,4 and in theoretical analysis of a mean-field model. We speculate that the scaling exponent could serve to classify combinatorial optimization problems into a small number of distinct categories, similar to universality classes in statistical physics.