Layers and Matroids for the Traveling Salesman's Paths

Frans Schalekamp (1) , András Sebő (2) , Vera Traub (3) , Anke van Zuylen (4)

1 Dpt of Computer Science [Cornell]
2 G-SCOP_OC - Optimisation Combinatoire
3 Universität Bonn = University of Bonn
4 WM - College of William and Mary [Williamsburg]

Gottschalk and Vygen proved that every solution of the well-known subtour elimination linear program for traveling salesman paths is a convex combination of a set of more and more restrictive "generalized Gao trees" of the underlying graph. In this paper we give a short proof of this, as a {\em layered} convex combination of bases of a sequence of more and more restrictive matroids. Our proof implies (via the matroid partition theorem) a strongly-polynomial combinatorial algorithm for finding this convex combination. This is a new connection of the TSP to matroids, offering also a new polyhedral insight.

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Frans Schalekamp, András Sebő, Vera Traub, Anke van Zuylen. Layers and Matroids for the Traveling Salesman's Paths. Operations Research Letters, 2018, 46, pp.60 - 63. ⟨10.1016/j.orl.2017.11.002⟩. ⟨hal-01592515⟩