Layers and Matroids for the Traveling Salesman's Paths

Abstract Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive “generalized Gao-trees”. We give a short proof of this fact, as a layered convex combination of bases of a sequence of increasingly restrictive matroids. A strongly polynomial, combinatorial algorithm follows for finding this convex combination, which is a new tool offering polyhedral insight, already instrumental in recent results for the s − t path TSP.

[1]  Jens Vygen,et al.  Better s-t-Tours by Gao Trees , 2016, IPCO.

[2]  Michel X. Goemans,et al.  Minimum Bounded Degree Spanning Trees , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[3]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[4]  Vera Traub,et al.  Approaching 3/2 for the s-t-path TSP , 2019, J. ACM.

[5]  David B. Shmoys,et al.  Improving christofides' algorithm for the s-t path TSP , 2011, STOC '12.

[6]  Jack Edmonds,et al.  Matroid Partition , 2010, 50 Years of Integer Programming.

[7]  William H. Cunningham,et al.  Testing membership in matroid polyhedra , 1984, J. Comb. Theory, Ser. B.

[8]  András Sebö,et al.  The Salesman's Improved Paths: A 3/2+1/34 Approximation , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).