Modular Forms in Combinatorial Optimization

Modular symmetries hidden in the combinatorial optimization framework remain mostly unexplored which has hindered any significant improvement in the solution quality. To unveil the modular structure, we map the cost and decision variables into complex domain and develop a novel framework for the Asymmetric Traveling Salesman Problem (ATSP). The transformed formulation is proven to be translation and inversion invariant, thereby allowing us to establish that achieving global optimum is equivalent to an infinite number of moment cancellations for each arc. The underlying idea is to achieve a delicate balance between cost and decision variables, expressed mathematically as an equilibrium condition, that allows for very strong modular symmetry to hold. The infinite moment cancellation is proven to be both necessary and sufficient condition for global optimality. In fact, we show that for strongly modular case, the rapid decay of moment contributions shall lead to series truncation with controllable error, allowing for efficient approximations. In contrast, weak modularity retains residual error thereby, reinforcing NP-hardness. These insights can inform the development of sophisticated algorithms that improve the quality of solutions.