On minimum reload cost paths, tours, and flows

Abstract

The concept of reload cost, that is of a cost incurred when two consecutive arcs along a path are of different types, naturally arises in a variety of applications related to transportation, telecommunication, and energy networks. Previous work on reload costs is devoted to the problem of finding a spanning tree of minimum reload cost diameter (with no arc costs) or of minimum reload cost. In this article, we investigate the complexity and approximability of the problems of finding optimum paths, tours, and flows under a general cost model including reload costs as well as regular arc costs. Some of these problems, such as shortest paths and minimum cost flows, turn out to be polynomially solvable while others, such as minimum shortest path tree and minimum unsplittable multicommodity flows, are NP-hard to approximate within any polynomial-time computable function

On the Quadratic Shortest Path Problem

“…We compare the results of the GLT, GGL, and IGL procedures with three other methods considered in [4]: the first is the root bound calculated by Cplex 12.4 when applied to the problem formulation (1). The other approaches, called QCR and OSU, are general approaches for solving quadratic 0-1 programming problems.…”

“…Therefore, only the quadratic terms of the form x ij x kl with j = k and i = l or with j = k and i = l have nonzero objective function coefficients. The AQSPP can be viewed as a generalization of the Reload Cost path introduced by Amaldi et al [1].…”

“…Recently, Amaldi et al [1] introduced new combinatorial optimization problems called reload cost paths, tours, and flows which have several applications in transportation networks, energy distribution networks, and telecommunication networks. In the reload cost problems, one is given a graph whose every edge is assigned a color and there is a reload cost when passing through a node on two edges that have different colors.…”

On Minimum Reload Cost Cycle Cover

“…The concept of reload costs was introduced in the seminal paper [6] where various applications are mentioned. This very natural concept, that models a kind of cost incurred during a transportation/transmission activity, has been, amazingly, considered only recently, so that the list of relevant references is comparatively small [2][3][5] [1]. We will be interested in asymmetric reload costs and in reload costs that satisfy the triangle inequality, where, for undirected graphs, we say that the reload costs satisfy the triangle inequality if, for any three edges e, e , e incident in a node of the graph, colored, respectively, with colors l, l , l , we have that c(l, l ) ≤ c(l, l ) + c(l , l ).…”

Reload cost trees and network design

“…She shows that unless P = N P the problem cannot be approximated within any constant α < 2 when reload costs are unrestricted and cannot be approximated within any constant β < 5/3 if the reload costs satisfy the triangle inequality. Amaldi et al [1] discuss the complexity of some path, tour, and flow problems under variable reload costs. Gourvès et al [11] focus on the complexity of the minimum reload cost s ‐ t path (which does not allow a node or edge to be revisited), trail (which allows a node to be revisited), and walk (which allows both a node and an edge to be revisited) problems under symmetric and asymmetric reload costs.…”