Generalizing and Unifying Gray-box Combinatorial Optimization Operators

TL;DR

Addresses how to design efficient gray-box operators across different combinatorial representations by introducing a unifying mathematical framework grounded in a decomposition theorem. A key result shows that for commuting, non-interacting moves the delta of a composed move equals the sum of the individual deltas, i.e., $\Delta_{h_1 \circ h_2} f(x) = \Delta_{h_1} f(x) + \Delta_{h_2} f(x)$ for all $x \in Y$, enabling constant-time hill-climbing and a unified partition crossover. Using Fourier transforms over finite groups, the framework identifies non-interacting moves, reinterprets existing operators like HBHC and PX, and guides the design of new gray-box operators for NP-hard problems such as the Linear Ordering Problem and the Single Machine Total Weighted Tardiness Problem. By being representation-agnostic and rooted in group theory, the approach promises faster development and redesign of problem-specific gray-box operators across diverse combinatorial domains.

Abstract

Gray-box optimization leverages the information available about the mathematical structure of an optimization problem to design efficient search operators. Efficient hill climbers and crossover operators have been proposed in the domain of pseudo-Boolean optimization and also in some permutation problems. However, there is no general rule on how to design these efficient operators in different representation domains. This paper proposes a general framework that encompasses all known gray-box operators for combinatorial optimization problems. The framework is general enough to shed light on the design of new efficient operators for new problems and representation domains. We also unify the proofs of efficiency for gray-box hill climbers and crossovers and show that the mathematical property explaining the speed-up of gray-box crossover operators, also explains the efficient identification of improving moves in gray-box hill climbers. We illustrate the power of the new framework by proposing an efficient hill climber and crossover for two related permutation problems: the Linear Ordering Problem and the Single Machine Total Weighted Tardiness Problem.