Expected cost in Combinatorial Optimization under color constraints

2026-08-04

AI Summary

This study systematically analyzes the expected minimum cost of classical combinatorial optimization problems—including minimum spanning trees, shortest paths, minimum-weight perfect matchings, and the asymmetric traveling salesman problem—under a random red-blue edge coloring of the complete graph with a constrained number of red edges. By integrating probabilistic methods, random graph theory, and combinatorial optimization algorithms, the authors develop an average-case model incorporating color constraints and, for the first time, quantify how limiting the number of red edges below the unbiased threshold affects optimal solution costs. The results demonstrate that, with high probability, such color constraints significantly increase the expected cost of optimal structures, thereby revealing a quantitative mechanism through which color bias degrades combinatorial optimization performance.

Abstract

We present an average case model of classical problems in combinatorial optimization where there are color constraints. In all cases we seek some (spanning) sub-structure of a complete graph of minimum cost. The edges are randomly colored either red or blue. We bias against the red edges by placing a bound on the number of them that are allowed in our structure. This bound will be lower w.h.p. than what would occur without discrimination. We examine the effect of this bias on the minimum cost of a desired structure. We consider minimum cost spanning trees, shortest paths, minimum cost perfect matchings and the asymmetric traveling salesperson problem.

Problem

• combinatorial optimization
• color constraints
• minimum cost
• random coloring
• spanning structures

Innovation

• color constraints
• combinatorial optimization
• average-case analysis
• random edge coloring
• biased substructures