Approximate Bi-Criteria Search by Efficient Representation of Subsets of the Pareto-Optimal Frontier

Abstract

We consider the bi-criteria shortest-path problem where we want to compute shortest paths on a graph that simultaneously balance two cost functions. While this problem has numerous applications, there is usually no path minimizing both cost functions simultaneously. Thus, we typically consider the set of paths where no path is strictly better than the others in both cost functions, a set called the Pareto-optimal frontier. Unfortunately, the size of this set may be exponential in the number of graph vertices and the general problem is NP-hard. While existing schemes to approximate this set exist, they may be slower than exact approaches when applied to relatively small instances and running them on graphs with even a moderate number of nodes is often impractical. The crux of the problem lies in how to efficiently approximate the Pareto-optimal frontier. Our key insight is that the Pareto-optimal frontier can be approximated using pairs of paths. This simple observation allows us to run a best-first-search while efficiently and effectively pruning away intermediate solutions in order to obtain an approximation of the Pareto frontier for any given approximation factor. We compared our approach with an adaptation of BOA*, the state-of-the-art algorithm for computing exact solutions to the bi-criteria shortest-path problem. Our experiments show that as the problem becomes harder, the speedup obtained becomes more pronounced. Specifically, on large roadmaps, when using an approximation factor of 10% we obtain a speedup on the average running time of more than X19.

A*pex: Efficient Approximate Multi-Objective Search on Graphs

“…If state s(π) is the goal state, then NAMOA* adds the path to the solution set and removes all paths from Open whose f -values are dominated by f (π). (Goldin and Salzman 2021), of merging path pairs ⟨π tl , π br ⟩ (orange) and ⟨π tl ′ , π br ′ ⟩ (blue) into path pair ⟨π tl new , π br new ⟩ (green).…”

“…For ε-bounded path pair ⟨π tl , π br ⟩, the g-value g(π tl ) = [g 1 (π tl ), g 2 (π tl )] of top-left path π tl is the lexicographically smallest g-value of all paths in this set, and the vector [g 2 (π br ), g 1 (π br )] of bottom-right path π br (called its reverse g-value) is the lexicographically smallest such vector of all paths in this set. Also, the g-value of the path pair weakly dominates the g-values of all paths in this set (because g 1 (π tl ) is the smallest g 1 -value of all paths in this set and g 2 (π br ) is the smallest g 2 -value of them), and the g-values of both the top-left and bottom-right paths ε-dominate the g-values of all paths in this set (Goldin and Salzman 2021).…”

“…Table 1 shows the average runtimes (in seconds) of PP-A*, A*pex-R, A*pex-L, and A*pex-G over all bi-objective road map instances that were solved by all four algorithms within the runtime limit and, in parentheses, the success rates (defined as the percentages of instances solved within the run-time limit). We omit the results for NAMOA*dr-ε because PP-A* significantly outperformed BOA*-ε on the instances in (Goldin and Salzman 2021), and BOA*-ε improves the efficiency of the dominance checks of NAMOA*dr-ε. The success rates of A*pex-R and A*pex-G were at least as large as the ones of A*pex-L for all approximation factors and road maps, and the success rates of A*pex-L were at least as large as the ones of PP-A* for all approximation factors and road maps and larger than the ones of PP-A* for ε ≤ 0.01 on road map NE.…”

“…Unfortunately, the size of the Pareto-optimal solution set can be exponential in the size of the graph being searched (Ehrgott 2005;Breugem, Dollevoet, and van den Heuvel 2017), which makes multi-objective search time-consuming. Researchers have therefore proposed to find an approximate Pareto-optimal solution set instead (Breugem, Dollevoet, and van den Heuvel 2017;Tsaggouris and Zaroliagis 2009;Warburton 1987;Goldin and Salzman 2021;Perny and Spanjaard 2008), that is, a set of paths such that any path in the Pareto-optimal solution set is ε-approximately dominated by some path in the approximate Pareto-optimal solution set. A path π ε-approximately dominates a path π ′ for some ε ≥ 0 iff each cost component of c(π) is no larger than (1 + ε) times the corresponding cost component of c(π ′ ), where ε is a user-provided approximation factor that can be different for different cost components.…”

Bounded-Cost Bi-Objective Heuristic Search

“…. Specifically, we follow the common practice in BOS and use the individual shortest path heuristic function (Hernández Ulloa et al 2020;Goldin and Salzman 2021;Pulido, Mandow, and Pérez-de-la Cruz 2015). That is, for h(n), h i (n) is the costminimal path from n to goal using the ith objective only.…”