The Graphical Traveling Salesperson Problem has no Integer Programming Formulation in the Original Space

TL;DR

This paper proves that the Graphical Traveling Salesperson Problem cannot be formulated as a mixed-integer program with only edge variables in the original space, highlighting fundamental limitations in modeling this problem.

Contribution

It establishes that no mixed-integer programming formulation with only edge variables in the original space exists for GTSP, strengthening previous complexity results.

Findings

No such MIP formulation exists at all.

The result is more rigorous than previous polynomial-time certification limitations.

The paper clarifies fundamental modeling constraints for GTSP.

Abstract

The Graphical Traveling Salesperson Problem (GTSP) is the problem of assigning, for a given weighted graph, a nonnegative number $x_e$ each edge $e$ such that the induced multi-subgraph is of minimum weight among those that are spanning, connected and Eulerian. Naturally, known mixed-integer programming formulations use integer variables $x_e$ in addition to others. Denis Naddef posed the challenge of finding a (reasonably simple) mixed-integer programming formulation that has integrality constraints only on these edge variables. Recently, Carr and Simonetti (IPCO 2021) showed that such a formulation cannot consist of polynomial-time certifyiable inequality classes unless $\mathsf{NP}=\mathsf{coNP}$. In this note we establish a more rigorous result, namely that no such MIP formulation exists at all.