Improving a constructive heuristic for the general routing problem

Abstract

The General Routing Problem (GRP) is a fundamental N P-hard vehicle routing problem, first defined by Orloff in 1974. It contains as special cases the Chinese Postman Problem, the Rural Postman Problem, the Graphical TSP and the Steiner TSP. We examine in detail a known constructive heuristic for the GRP, due to Christofides and others. We show how to speed it up, in both theory and practice, while obtaining solutions that are at least as good. Computational results show that, for large instances, our implementation is faster than the original by several orders of magnitude.

Fast upper and lower bounds for a large‐scale real‐world arc routing problem

“…In [12], we show that one can use Euclidean distances instead of real road distances when solving certain node routing problems, while incurring only a small loss of quality. In [13], we present a method, called sparsification, for improving solutions to another vehicle routing problem. We will adapt both of these ideas to our problem (see Subsections 3.1-3.4).…”

“…The costs are also indicated on the links. Suppose that the first two phases have produced a giant tour which traverses the required links in the order (1, 2), (3,4), (8,9), (9,10), (13,12), (12,11). Figure 1(b) shows the corresponding digraph G ′ , before we remove any arcs from A 2 .…”

“…In the fourth phase of our heuristic, we attempt to reduce the length of the giant tour. The procedure is essentially an extension of the “sparsification” method in [13] to the case of mixed graphs. The procedure is rather complicated, but we have found that it is very worthwhile, typically leading to reductions in tour length of over 10% in just a few seconds.…”