Analysis of FPTASes for the multi-objective shortest path problem

Abstract

We propose a new FPTAS for the multi-objective shortest path problem. The algorithm uses elements from both an exact labeling algorithm and an FPTAS proposed by Tsaggouris and Zaroliagis (2009). We analyze the running times of these three algorithms both from a theoretical and a computational point of view. Theoretically, we show that there are instances for which the new FPTAS runs an arbitrary times faster than the other two algorithms. Furthermore, for the bi-objective case, the number of approximate solutions generated by the proposed FPTAS is at most the number of Pareto-optimal solutions multiplied by the number of nodes. By performing a set of computational tests, we show that the new FPTAS performs best in terms of running time in case there are many dominated paths and the number of Pareto-optimal solutions is not too small.

Approximating Multiobjective Shortest Path in Practice

“…None of the above strategies seem to have been implemented or tested until the publication of Breugem, Dollevoet and van den Heuvel [5]. They introduce the idea of a cell decomposition by Papadimitriou and Yannakakis to a traditional label-setting algorithm.…”

“…• Compared to the algorithm by Warburton as well as Papadimitriou and Yannakakis, we only consider one pass of a labeling algorithm. As noted by [25] and [5], these passes are the most expensive operations of the algorithm.…”

“…In practice, these approximation algorithm do not seem to play a role at all. To the best of our knowledge, the only practical study where approximation algorithms for MOSP have been implemented is very recent from Breugem, Dollevoet and van den Heuvel [5] (see below). Still, only very small artificial test instances with up to 20 nodes have been considered in their work.…”

Simple paths with exact and forbidden lengths

“…The problem E xact P ath () can also be formulated as a special case of a constrained path problem , which is to maximize path length, subject to the constraint that the path length does not exceed a certain value, or to minimize the path length, subject to the constraint that the path length is at least a certain value. There exists a voluminous literature on the constrained and bi‐objective path problems, see, for example, Joksch (), Dial (), Hansen (), Aneja, Aggarwal, and Nair (), Desrochers (), Warburton (), Hassin (), Lorenz and Raz (), Ergun, Sinha, and Zhang (), Righini and Salani (), Boland, Dethridge, and Dumitrescu (), Garcia (), Tsaggouris and Zaroliagis (), and Breugem, Dollevoet, and van den Heuvel (). We will present new observations about the relations between the problem E xact P ath () and other path problems with the forbidden gaps.…”

Efficient index structures for reachability and shortest path queries

“…α-CSP without index. The current state-of-the-art solution for α-CSP is Breugem's method [20] (called CP-Dijk in the following), which resembles Sky-Dijk with a conservative pruning (CP) technique originally developed by Tsaggouris and Zaroliagis [90]. Specifically, given an α-CSP with origin s, destination t, cost constraint θ, and approximation ratio α, CP-Dijk applies the same data structures and follows the same steps as Sky-Dijk, with a single modification: that each label set L(v) maintains the set of paths that are not n √ α-dominated (Definition 5.2) by another path in L(v), where n is the total number of vertices in the input graph G. Because n √ α-dominance is a relaxed condition of exact (i.e., 1-) dominance, this modification leads to faster query processing.…”