The Windy General Routing Polyhedron: A Global View of Many Known Arc Routing Polyhedra

Abstract

The Windy Postman Problem consists of finding a minimum cost traversal of all the edges of an undirected graph with two costs associated with each edge, representing the costs of traversing it in each direction. In this paper we deal with the Windy General Routing Problem (WGRP), in which only a subset of edges must be traversed and a subset of vertices must be visited. This is also an NP-hard problem that generalizes many important Arc Routing Problems (ARP's) and has some interesting real-life applications. Here we study the description of the WGRP polyhedron, for which some general properties and some large families of facet-inducing inequalities are presented. Moreover, since the WGRP contains many well-known routing problems as special cases, this paper also provides a global view of their associated polyhedra. Finally, for the first time, some polyhedral results for several ARP's defined on mixed graphs formulated using two variables per edge are presented.

Min‐Max K‐vehicles windy rural postman problem

“…We will call aggregate inequalities to these inequalities. R-odd cut, K-C, Honeycomb and other inequalities for the WRPP described in Corberán et al [11] are then valid inequalities for the MM K-WRPP. In particular, the aggregate R-odd cut inequalities are: (8) and are based on the fact that any K-WRPP solution must cross any given edge cutset an even number of times.…”

“…It is known ( [11]) that, if G is connected, dim(WRPP(G))= 2|E|−|V | + 1. Let us call m = dim(WRPP(G)).…”

“…is not a bridge of G then inequality x ij ≥ 0 is facet-inducing for WRPP(G) (see [11]) and there are m = dim(WRPP(G))= 2|E|−|V | + 1 affinely independent WRPP tours on G satisfying x ij = 0, say w 1 , w 2 , . .…”

“…We define a WRPP instance G in which all the edges in E are required except those in δ(S). Given that subgraphs G(S) and G(V \S) are connected, inequality x(δ(S)) ≥ 2 is facet-inducing for WRPP(G ) (see [11]) and there exist m linearly independent tours w 1 , w 2 , . .…”

“…Proof: Given that G(S) and G(V \S) are connected, the R-odd cut inequality x(δ(S)) ≥ |δ R (S)| + 1 is facet-inducing for WRPP(G) ( [11]) and there are m linearly independent tours w 1 , w 2 , . .…”