The Travelling Salesman Polytope and {0, 2}-Matchings

Abstract

A {0, 2}-matching is an assignment of the integers 0, 2 to the edges of a graph G such that for every node the sum of the integers on the incident edges is at most two. A tour is the (0-1)-incidence vector of a hamilton cycle. We study the polytope P ( G ), defined to be the convex hull of the {0, 2}-matchings and tours of G . When G has an odd number of nodes, the travelling salesman polytope, the convex hull of the tours, is a face of P ( G ). We obtain the following results: (i) We completely characterize those facets of P ( G ) which can be induced by an inequality with (0-1)-coefficients. (ii) We prove necessary properties for any other facet inducing inequality, and exhibit a class of such inequalities with the property that for any pair of consecutive positive integers, there exists an inequality in our class whose coefficients include these integers. (iii) We relate the facets of P ( G ) to the facets of the travelling salesman polytope. In particular, we show that for any facet F of the travelling salesman polytope, there is a unique facet of P ( G ) whose intersection with the travelling salesman polytope is exactly F .