Postman Problems on Mixed Graphs

Abstract

The <i>mixed postman problem</i> consists of finding a minimum cost tour of a mixed graph <i>M</i> = (<i>V</i>,<i>E</i>,<i>A</i>) traversing all its edges and arcs at least once. We prove that two well-known linear programming relaxations of this problem are equivalent. The <i>extra cost</i> of a mixed postman tour <i>T</i> is the cost of <i>T</i> minus the cost of the edges and arcs of <i>M</i>. We prove that it is <i>NP</i>-hard to approximate the minimum extra cost of a mixed postman tour. A related problem, known as the <i>windy postman problem</i>, consists of finding a minimum cost tour of an undirected graph <i>G</i>=(<i>V</i>,<i>E</i>) traversing all its edges at least once, where the cost of an edge depends on the direction of traversal. We say that <i>G</i> is <i>windy postman perfect</i> if a certain <i>windy postman polyhedron O</i> (<i>G</i>) is integral. We prove that series-parallel undirected graphs are windy postman perfect, therefore solving a conjecture of Win. Given a mixed graph <i>M</i> = (<i>V</i>,<i>E</i>,<i>A</i>) and a subset <i>R</i> &#8838; <i>E</i> &#8746; <i>A</i>, we say that a mixed postman tour of <i>M</i> is <i>restricted</i> if it traverses the elements of <i>R</i> exactly once. The <i>restricted mixed postman problem</i> consists of finding a minimum cost restricted tour. We prove that this problem is <i>NP</i>-hard even if <i>R</i>=<i>A</i> and we restrict <i>M</i> to be planar, hence solving a conjecture of Veerasamy. We also prove that it is <i>NP</i>-complete to decide whether there exists a restricted tour even if <i>R</i>=<i>E</i> and we restrict <i>M</i> to be planar. The <i>edges postman problem</i> is the special case of the restricted mixed postman problem when <i>R</i>=<i>A</i>. We give a new class of valid inequalities for this problem. We introduce a relaxation of this problem, called the <i>b-join problem</i>, that can be solved in polynomial time. We give an algorithm which is simultaneously a 4/3-approximation algorithm for the edges postman problem, and a 2-approximation algorithm for the extra cost of a tour. The <i>arcs postman problem</i> is the special case of the restricted mixed postman problem when <i>R</i>=<i>E</i>. We introduce a class of necessary conditions for <i>M</i> to have an arcs postman tour, and we give a polynomial-time algorithm to decide whether one of these conditions holds. We give linear programming formulations of this problem for mixed graphs arising from windy postman perfect graphs, and mixed graphs whose arcs form a forest.