Algebraic Structure Discovery for Real World Combinatorial Optimisation Problems: A General Framework from Abstract Algebra to Quotient Space Learning

TL;DR

This paper introduces a general algebraic framework that uncovers hidden structures in combinatorial optimization problems, enabling more efficient search by reducing the search space through quotient spaces.

Contribution

It formalizes a method to identify algebraic structures, construct quotient spaces, and optimize over these reduced spaces, demonstrated on rule-based tasks and clinical data.

Findings

Quotient-space-aware genetic algorithms outperform standard methods in recovering global optima.

The framework applies to rule-combination tasks like patient subgroup discovery.

Exposing algebraic structure improves efficiency and diversity in optimization.

Abstract

Many combinatorial optimisation problems hide algebraic structures that, once exposed, shrink the search space and improve the chance of finding the global optimal solution. We present a general framework that (i) identifies algebraic structure, (ii) formalises operations, (iii) constructs quotient spaces that collapse redundant representations, and (iv) optimises directly over these reduced spaces. Across a broad family of rule-combination tasks (e.g., patient subgroup discovery and rule-based molecular screening), conjunctive rules form a monoid. Via a characteristic-vector encoding, we prove an isomorphism to the Boolean hypercube $\{0,1\}^n$ with bitwise OR, so logical AND in rules becomes bitwise OR in the encoding. This yields a principled quotient-space formulation that groups functionally equivalent rules and guides structure-aware search. On real clinical data and synthetic benchmarks, quotient-space-aware genetic algorithms recover the global optimum in 48% to 77% of runs versus 35% to 37% for standard approaches, while maintaining diversity across equivalence classes. These results show that exposing and exploiting algebraic structure offers a simple, general route to more efficient combinatorial optimisation.