Multicriteria path and tree problems: discussion on exact algorithms and applications

Abstract

Multicriteria/multiobjective path and tree models are useful in many applications. Particularly, in Internet routing problems they seem to lead to promising approaches. In the first part of this paper, we classify and present the main exact approaches dealing with several multicriteria path problems putting in evidence the shortest path problem. In the second part, we review exact algorithms dedicated to some multicriteria tree problems, namely the minimum spanning tree and the minimum cost/minimum label spanning tree problems. Finally, the application of these models is exemplified. the emergence of multimedia applications such as audio, video services and video conferencing. The relevant criteria for many problems in this area are not only multifaceted but commonly of an heterogeneous nature, for example economic, technical and socio-economic criteria. These issues mean that in many situations the models for decision support in these areas become more realistic if different aspects are explicitly considered by building a consistent set of criteria rather than just aggregating some of them a priori in a single function and transforming the others into constraints as has been done in earlier OR models dedicated to problems of this field. Multicriteria models enable deeper analysis -for instance, evaluating trade-offs among the distinct performance metrics in a mathematically consistent manner when such metrics are conflicting objectives -helping the decision makers (DM) to look for satisfactory compromise solution(s).Note that even in the cases where an a priori aggregation of criteria is required (for instance, in online automatic decision making situations), an explicit multicriteria modeling has the advantage of enabling a deeper insight, regarding the prosecution of several problem issues.In the second section of this paper we classify and make an overview of multicriteria path problems. Taking into account its importance, we pay special attention to the shortest path problem. We make an overview of exact algorithms dedicated to multicriteria path problems and we also put in evidence some relevant theoretical results.Note that considering explicitly multiple and conflicting criteria, a global optimum does not exist any more. Here the concept of optimum is substituted by the concept of efficient/nondominated solution (a path in this case). An efficient (also known in the literature as non-inferior or Pareto optimal) path is a feasible path such that there are no other feasible paths that can improve the value of one objective function without worsening the value of at least one of the other objective functions. The mapping of an efficient path in the objective functions space leads to the corresponding non-dominated solution/point. Furthermore, these solutions can be supported and unsupported non-dominated solutions. While supported non-dominated solutions are non-dominated solutions located on the boundary of the convex hull of the feasible set in the objective function space, unsupported non-dominated soluti...

An exact method for constructing minimal cost/minimal SRLG spanning trees over optical networks

“…We developed an exact resolution approach for the formulated bicriteria spanning tree problem, based on an extension of the algorithm proposed in [46] for the minimal cost/minimal label spanning tree problem. Suggestions on possible applications of minimum label spanning tree problems to communication networks were outlined in [47] and [40]. Note that, while the minimal cost spanning tree problem (MCST) can be solved in polynomial time by using, for example, the classical algorithms by Kruskal [48] or by Prim [49], the minimum label spanning tree problem (which seeks to determine a spanning tree with the minimal number of different labels, assuming that each edge of the network is associated with a label) introduced in [50], was proven in this work to be NP-hard.…”

“…A survey on multicriteria minimum spanning tree problems, presenting theoretical results and algorithms, is in [39]. A review on multicriteria path and tree problems including a discussing on exact algorithms and applications is presented by Clímaco and Pascoal [40]. A proposal of a generic conceptual framework for the development of consistently multicriteria routing models in IP/QoS networks is described in [41].…”

Path Planning and Vehicle Scheduling Optimization for Logistic Distribution of Hazardous Materials in Full Container Load

“…Generally, there are two types of solutions to it: dynamic programming (DP) [8,9] and ranking [10]. Pulse algorithm [11] is a precise algorithm to solve BSP.…”

Dimensionality reduction in multiobjective shortest path search

“…These include enumerative approaches (label-setting and label-correcting), ranking algorithms, and two-phase algorithms (e.g. see [4,2,34,35,12,37,31,5]). For the one-to-one problem, where single start and goal nodes are designated in the graph, label-setting algorithms are generally the best option for arbitrary start-goal pairs in large graphs, e.g.…”

“…Multiobjective (MO) shortest path problems arise in many fields, such as vehicle path planning [17,39,40], urban transportation networks [6,9], robot surveillance [7], satellite scheduling [11], routing in telecommunication networks [5], and route planning in different contexts [1,3,8,16].…”