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

06/18/2021

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by   Matthias Walter, et al.

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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 𝖭𝖯=𝖼𝗈𝖭𝖯. In this note we establish a more rigorous result, namely that no such MIP formulation exists at all.