A Surface-Based Formulation of the Traveling Salesman Problem

TL;DR

This paper introduces a novel surface-based MILP formulation for the symmetric TSP, replacing edge selection with triangle-based surface construction, offering a new perspective and heuristic approach.

Contribution

It presents an exact surface-building formulation for TSP using Euler characteristic constraints, providing a new geometric perspective and a practical heuristic.

Findings

Exact when using all triangles but computationally intensive

Offers a compact heuristic with sparse triangle sets

Replaces subtour elimination with surface boundary constraints

Abstract

We present an exact formulation of the symmetric Traveling Salesman Problem (TSP) that replaces the classical edge-selection view with a surface-building approach. Instead of selecting edges to form a cycle, the model selects a set of connected triangles where the boundary of the resulting surface forms the tour. This method yields a mixed-integer linear programming (MILP) formulation where a tree constraint enforces global connectivity, while local connectivity at each vertex is guaranteed via Euler characteristic constraints, replacing the need for subtour elimination. The formulation is exact when applied to the complete set of all triangles, despite being computationally intractable for all but the smallest instances. In practice, it provides a compact and effective heuristic when restricted to a sparse candidate set such as Delaunay triangulation.