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

arXiv:2604.04941v1 Announce Type: new
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 $,1^n$ 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.