Layers and Matroids for the Traveling Salesman's Paths

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 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 which was instrumental in the most recent approximation algorithm and bound on the integrality ratio for the s− t path TSP. keywords: path traveling salesman problem (TSP), matroid partition, approximation algorithm, spanning tree, Christofides’ heuristic, T -join, polyhedra