Series–parallel graphs are windy postman perfect

Abstract

The windy postman problem is the NP-hard problem of finding the minimum cost of a tour traversing all edges of an undirected graph, where the cost of an edge depends on the direction of traversal. Given an undirected graph G, we consider the polyhedron O ( G ) induced by a linear programming relaxation of the windy postman problem. We say that G is windy postman perfect if O ( G ) is integral. There exists a polynomial-time algorithm, based on the ellipsoid method, to solve the windy postman problem for the class of windy postman perfect graphs. By considering a family of polyhedra related to O ( G ) , we prove that series–parallel graphs are windy postman perfect, therefore solving a conjecture of Win.