A constructive heuristic for the Undirected Rural Postman Problem

Abstract

This paper describes a constructive heuristic for the well-known Undirected Rural Postman Problem. At each iteration, the procedure inserts a connected component of the required edges and performs a local postoptimization. Computational results on a set of benchmark instances with up to 350 vertices show that the proposed procedure is competitive with the classical Frederickson procedure. Its use is recommended when a high-quality solution is needed in a short amount of time (e.g., in laser plotter applications).

Introduction

The aim of this paper is to present a constructive heuristics for the Undirected Rural Postman Problem (RPP) defined as follows. Let $G=(V,E)$ be an undirected graph, where V is the vertex set, E is the edge set, $c_{\mathit{ij}}(⩾0)$ is the cost of traversing edge $(v_i,v_j)\in E$, $d_{\mathit{ij}}$ is the cost of a shortest chain $l_{\mathit{ij}}$ between $v_i$ and $v_j$, and $R\subseteq E$ is a set of required edges. The RPP amounts to determining a least cost tour traversing each edge of R at least once. Equivalently, solving the RPP is to determine a least cost set of additional edges that, along with the required edges, makes up a Eulerian and connected subgraph.

Applications of the RPP to the control of plotting and drilling machines [1] and to the optimization of laser-plotter beam movements [2] have been described in recent years. Furthermore, the RPP is the unconstrained version of more general classes of multi-vehicle Capacitated Arc Routing Problems arising, for instance, in garbage collection, road gritting, mail delivery, network maintenance, etc. [3], [4], [5].

Exact algorithms for the RPP have been developed by Christofides et al. [6], Corberán and Sanchis [7], Letchford [8], Ghiani and Laporte [9] and Fernandez et al. [10]. In addition, constructive heuristics for the RPP have been presented by Frederickson [11], Pearn and Wu [12], while improvement procedures have been described by Hertz et al. [13].

We illustrate a novel heuristic which iteratively inserts a connected component of the required edges into the partial solution, while performing a local re-optimization. The remainder of the paper is organized as follows. Section 2 depicts the new procedure, followed by the computational results in Section 3 and by the conclusions in Section 4.