A Modular-Form Framework for Global Optimality in the Asymmetric Traveling-Salesman Problem

TL;DR

This paper introduces a novel mathematical framework linking the asymmetric traveling salesman problem to holomorphic cusp forms, providing a new approach to identify global optima through number theory techniques.

Contribution

It develops an alternative formulation of ATSP using cusp forms and introduces a three-step filtering process for certifying global optimality.

Findings

Holomorphic cusp forms vanish at the global optimum under certain conditions.

A three-step filter using Fourier coefficients, Hecke recursions, and L-function parity tests certifies optimality.

The framework suggests a new connection between discrete optimization and number theory.

Abstract

In this paper, we develop an alternate formulation of Asymmetric Traveling Salesman Problem (ATSP). The equivalent problem is to find the zeros of a holomorphic cusp form on the principal congruence subgroup, $\Gamma(4) $. The resultant Poincar{\'e} series gives a cusp form whose interior zeros are in bijection with the arc that constitute optimal Hamiltonian cycle. We show that for any weight, $\ell$ and number of directed arcs, $|A|$ such that $4\ell-7<2|A| $, the holomorphic cusp form vanishes at global optimum. Furthermore, a three step filter consisting of Fourier coefficients, Hecke recursions and completed $L$-function parity test provides a scalar certificate for global optimality. The framework is a potential bridge between discrete optimization and number theory suggesting an alternate view on complexity theory.