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

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.


Introduction
Let G = (V, E) be a graph and let c ∈ R E . 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].
x e is even ∀v ∈ V (1c) Here, δ(S) := {e ∈ E : |e ∩ S| = 1} and δ(v) := δ({v}) denote the cuts induced by node set S ⊆ V and node v ∈ V , respectively. For each edge e ∈ E, the variable x e indicates how often e is traversed in the tour.
Among these were path, wheelbarrow and bicycle inequalities. In order to turn (1) into a mixed-integer programming model (MIP), constraint (1c) can be replaced by this pair of constraints: These additional y-variables are artificial and their presence has no impact on the linear programming relaxation of (1). For this reason, Naddef posed the challenge of finding a simple mixed-integer programming formulation that involves, apart from the x-variables, only continuous variables [4]. According to [1], he had the formulation from [2] with path, wheelbarrow and bicycle inequalities in mind. There exist other (mixed-)integer programming formulations for the GTSP, see [1,3], all of which requiring additional integral variables.
Recently, Carr and Simonetti showed that such a formulation cannot be nice in the sense that it cannot consist of inequality families for which one can certify membership in polynomial time, provided NP = coNP (see Section 4.2 in [1]). The purpose of this paper is to show that the reason for the non-existence of a simple formulation does not lie in complexity theory. In fact, we show that no such formulation exists at all: Theorem 1. The GTSP has no mixed-integer programming formulation whose only integer variables are the x-variables from (1). Proof of Theorem 1. We consider the graph G ⋆ = (V ⋆ , E ⋆ ) from Fig. 1 with unit edge costs c ⋆ ∈ R E ⋆ . It is easy to see that G ⋆ has no Hamiltonian cycle, and therefore there is no solution of value |V ⋆ | = 5. Hence, the tours in Figure (1
Assume, for the sake contradiction, that there exists a mixed-integer programming formulation Q = {(x, y) ∈ Z E × R q : Ax + By ≤ d} that has integrality constraints only for the x-variables. Hence, the projection of Q onto the x-variables is the set of feasible solutions to (1), and hence is equivalent to (1). In particular, feasibility of x ⋆ 1 and x ⋆ 2 for (1) implies that there exist y ⋆ 1 , y ⋆ 2 ∈ R q such that (x ⋆ i , y ⋆ i ) ∈ Q for i = 1, 2. Now let y ⋆ ∈ R q be the midpoint of y ⋆ 1 and y ⋆ 2 . By convexity of the linear relaxation of Q and integrality of x ⋆ , also (x ⋆ , y ⋆ ) is an optimal solution to (3). This contradicts the fact that x ⋆ is infeasible for (1).