Privatized rural postman problems

In this work we analyze the privatized rural postman problem which is the edge version of the traveling salesman problems with profits. The problem is defined on a undirected graph G(V, E) with a distinguished vertex d, called the Depot. There are two non-negative real functions on the edge set E, which define the value of a cycle in G: one is the profit function, b, and the other one is the cost function, c. They have different meanings when a cycle C traverses an edge e (possibly more than once), because we pay a cost c(e) every time e is traversed, but we collect the profit be only the first time e is traversed. The privatized rural postman problem is to find a cycle C*, passing through d and not necessarily simple, which maximizes the sum of the values of the edges traversed in C*. That is, max(C) {Sigma(e epsilon C)(b(e)-t(e)c(e))} where t(e) is the number of times that edge e is traversed in C. We study some properties of the problem: we show that it is NP-hard, its relation with known and new problems, and special cases with good algorithms. We also analyze several integer linear systems of inequalities, which define the polyhedral structure of the problem, and we give dominance and preprocessing conditions. We finish with some remarks and comments about future research. (c) 2005 Elsevier Ltd. All rights reserved.