The Graphical Traveling Salesperson Problem has no integer programming formulation in the original space

Matthias Walter

doi:10.1016/j.orl.2021.06.015

Abstract

The Graphical Traveling Salesperson Problem is the problem of assigning, for a given weighted graph, a nonnegative number xe to each edge e such that the induced multi-subgraph is of minimum weight among those that are spanning, connected and Eulerian. Known MIP formulations are based on integer variables xe. Carr and Simonetti (IPCO 2021) showed that unless NP=coNP, no (reasonable) formulation can have integrality constraints only on x-variables, a challenge posed by Denis Naddef. We establish the same result unconditionally.

Highlights

Let G = (V, E) be a graph and let c ∈ RE
The Graphical Traveling Salesperson problem is about finding c-minimum cost tour in G that visits each node at least once, where edges can be used multiple times
The authors of [2] describe several classes of inequalities that are valid for the GTSP polyhedron P gtsp(G) defined as the convex hull of all feasible solutions, i.e., P gtsp(G) := conv{x ∈ ZE : x satisfies (1b), (1c) and (1d)}

Summary

Introduction

The Graphical Traveling Salesperson problem is about finding c-minimum cost tour in G that visits each node at least once, where edges can be used multiple times. It can be formulated as the following constraint integer program due to Cornéjols, Fonlupt and Naddef [2]. For each edge e ∈ E, the variable xe indicates how often e is traversed in the tour.

Objectives

Results

Conclusion