Extend the gray-box framework to non-commutative moves

Extend the decomposition-theorem-based gray-box optimization framework for efficient hill climbers and partition crossover to sets of moves that do not commute under composition, and determine the conditions under which the delta function remains separable when applying non-commutative move families in combinatorial optimization.

Background

The framework introduced in the paper proves a Decomposition Theorem that enables constant-time hill climbing and optimal partition crossover by requiring families of moves that commute under composition and are pairwise non-interacting. This assumption holds in abelian settings such as binary strings (Z2^n) and specific permutation neighborhoods.

The authors explicitly flag as an open question the extension of this framework beyond commutative moves. Addressing non-commutative move families would broaden applicability to more general permutation operations and other non-abelian group actions, but it requires establishing new conditions ensuring delta separability without the commutativity assumption.