José M. Sanchis

A polyhedral approach to the Rural Postman Problem

Abstract In this paper we study the polyhedron associated with the Rural Postman Problem (RPP). Because the RPP is NP-hard, we cannot expect to find a complete description of the rural postman polyhedron of a general graph, but a partial knowledge of such a description frequently proves to be useful for both theoretical and computational purposes. We have tried to characterize the facial structure of this unbounded full-dimensional polyhedron. Sets of valid inequalities inducing facets have been studied as well as their use in a cutting-plane algorithm. The application of this algorithm to a set of RPP instances taken from the literature and two instances of larger size taken from a real world graph is described. All these instances were solved to optimality.

A cutting plane algorithm for the General Routing Problem

Abstract.The General Routing Problem (GRP) is the problem of finding a minimum cost route for a single vehicle, subject to the condition that the vehicle visits certain vertices and edges of a network. It contains the Rural Postman Problem, Chinese Postman Problem and Graphical Travelling Salesman Problem as special cases. We describe a cutting plane algorithm for the GRP based on facet-inducing inequalities and show that it is capable of providing very strong lower bounds and, in most cases, optimal solutions.

Linear Programming Based Methods for Solving Arc Routing Problems

From the pioneering works of Dantzig, Edmonds and others, polyhedral (i.e. linear programming based) methods have been successfully applied to the resolution of many combinatorial optimization problems. See Junger, Reinelt & Rinaldi (1995) for an excellent survey on this topic. Roughly speaking, the method consists of trying to formulate the problem as a Linear Program and using the existing powerful methods of Linear Programming to solve it.

Lower bounds and heuristics for the Windy Rural Postman Problem

In this paper we present several heuristic algorithms and a cutting-plane algorithm for the Windy Rural Postman Problem. This problem contains several important Arc Routing Problems as special cases and has very interesting real-life applications. Extensive computational experiments over different sets of instances are also presented.

New Results on the Mixed General Routing Problem

In this paper, we deal with the polyhedral description and the resolution of the Mixed General Routing Problem. This problem, in which the service activity occurs both at some of the nodes and at some of the arcs and edges of a mixed graph, contains a large number of important arc and node routing problems as special cases. Here, a large family of facet-defining inequalities, the Honeycomb inequalities, is described. Furthermore, a cutting-plane algorithm for this problem that incorporates new separation procedures for the K-C, Regular Path-Bridge, and Honeycomb inequalities is presented. Branch and bound is invoked when the final solution of the cutting-plane procedure is fractional. Extensive computational experiments over different sets of instances are included.

A heuristic algorithm based on Monte Carlo methods for the rural postman problem

Abstract In Routing Problems the aim is to determine a minimum cost traversal over a graph satisfying some specified constraints. Most of them are NP-hard problems and many different heuristic solution algorithms have been proposed. The name Monte Carlo, MC, applies to a set of heuristic procedures with the common feature of using random numbers to simulate a given process. MC approach has not been applied to the framework of Routing Problems in the literature. The purpose of this paper is to demonstrate that MC methods could be useful in implementing heuristic algorithms for Routing Problems. In particular, we design an efficient MC heuristic algorithm for the well known Rural Postman Problem (RPP), for which we have a set of instances with known optimal solution taken from the literature. The Rural Postman Problem (RPP) consists of finding a minimum cost traversal of a specified arc subset of a graph. Given that the RPP is a NP-hard problem, heuristic algorithms are interesting both to handle large size instances and to provide upper bounds that could be used in branch and cut procedures. In this paper we propose a heuristic algorithm for the RPP based on Monte Carlo methods. We simulate a vehicle travelling randomly over the graph, jumping from one node to another on the basis of certain probabilities. Monte Carlo methods provide a simple approach to many different Routing Problems and they are easily implemented in a computer code. The application of this algorithm to a set of RPP instances taken from the literature demonstrates that, using the appropriate probabilities, they are also efficient.

An asymmetric TSP with time windows and with time-dependent travel times and costs: An exact solution through a graph transformation

In this paper we deal with an extended version of the Asymmetric Traveling Salesman Problem with Time Windows (ATSPTW) that considers time-dependent travel times and costs, for a more accurate approximation of some routing problems inside large cities, in which the time or cost of traversing certain streets (e.g. main avenues) depends on the moment of the day (for example rush-hours). Unlike other existing papers about time-dependent routing problems, we focus on an exact method for solving this new problem. For this end we first transform the problem into an Asymmetric Generalized TSP and then into a Graphical Asymmetric TSP. In this way, we can apply a known exact algorithm for the Mixed General Routing Problem, which seems to run well with our resulting instances. Computational results are presented on a set of 270 adapted instances from benchmark ATSPTW instances.

The mixed general routing polyhedron

Abstract. In Arc Routing Problems, ARPs, the aim is to find on a graph a minimum cost traversal satisfying some conditions related to the links of the graph. Due to restrictions to traverse some streets in a specified way, most applications of ARPs must be modeled with a mixed graph. Although several exact algorithms have been proposed, no polyhedral investigations have been done for ARPs on a mixed graph. In this paper we deal with the Mixed General Routing Problem which consists of finding a minimum cost traversal of a given link subset and a given vertex subset of a mixed graph. A formulation is given that uses only one variable for each link (edge or arc) of the graph. Some properties of the associated polyhedron and some large families of facet-inducing inequalities are described. A preliminary cutting-plane algorithm has produced very good lower bounds over a set of 100 randomly generated instances of the Mixed Rural Postman Problem. Finally, applications of this study to other known routing problems are described.

A GRASP heuristic for the mixed Chinese postman problem

Abstract Arc routing problems (ARPs) consist of finding a traversal on a graph satisfying some conditions related to the links of the graph. In the Chinese postman problem (CPP) the aim is to find a minimum cost tour (closed walk) traversing all the links of the graph at least once. Both the Undirected CPP, where all the links are edges that can be traversed in both ways, and the Directed CPP, where all the links are arcs that must be traversed in a specified way, are known to be polynomially solvable. However, if we deal with a mixed graph (having edges and arcs), the problem turns out to be NP -hard. In this paper, we present a heuristic algorithm for this problem, the so-called Mixed CPP (MCPP), based on greedy randomized adaptive search procedure (GRASP) techniques. The algorithm has been tested and compared with other known and recent methods from the literature on a wide collection of randomly generated instances, with up to 200 nodes and 600 links, producing encouraging computational results. As far as we know, this is the best heuristic algorithm for the MCPP, with respect to solution quality, published up to now.

The general routing problem polyhedron: Facets from the RPP and GTSP polyhedra

In this paper we study the polyhedron associated with the General Routing Problem (GRP). This problem, first introduced by Orloff in 1974, is a generalization of both the Rural Postman Problem (RPP) and the Graphical Traveling Salesman Problem (GTSP) and, thus, is NP -hard. We describe a formulation of the problem such that from every non-trivial facet-inducing inequality for the RPP and GTSP polyhedra, we obtain facet-inducing inequalities for the GRP polyhedron. We describe a new family of facet-inducing inequalities for the GRP, the honeycomb constraints, which seem to be very useful for solving GRP and RPP instances. Finally, new classes of facets obtained by composition of facet-inducing inequalities are presented.