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

TL;DR

The paper reframes the asymmetric traveling salesman problem (ATSP) as a modular-form problem by mapping tours to zeros of a cusp form on $\Gamma(4)$ via a Poincaré lift. For a chosen even weight $\ell$, the lift yields a cusp form $F_{\ell} \in S_{\ell}(\Gamma(4))$ whose interior zeros correspond to in-tour arcs, and when $4\ell-7<2|A|$, these zeros force $F_{\ell}\equiv0$ at the global optimum. Global optimality is certified by a three-filter pipeline—Fourier-coefficient vanishing via a Vandermonde system, Hecke-relation constraints, and a central-value parity test of the completed $L$-function $\Lambda(F_{\ell},s)$—which together yield a scalar certificate. Overall, the work proposes a conceptual bridge between discrete optimization and number theory, with potential implications for understanding problem hardness through arithmetic structure.

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, $Γ(4) $. The resultant Poincar{é} 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.

Theorems & Definitions (30)

• Definition 1: imaginary axis
• Lemma 1
• proof
• Remark 1
• Remark 2
• Lemma 2
• proof
• proposition 1: Simple zeros of $F_\ell$ imply optimal Hamiltonian tour
• proof
• Theorem 1: Optimal interior zeros of $F_\ell$