Improved Approximation Schemes for the Restricted Shortest Path Problem

TL;DR

This paper develops improved approximation algorithms for the NP-hard Restricted Shortest Path problem, achieving near-optimal solutions efficiently on various graph classes.

Contribution

It introduces simpler, more efficient approximation schemes for RSP on multiple graph classes, improving upon existing algorithms.

Findings

Achieves (1 + ε)-approximations in O(mn/ε) time for certain graph classes.

Matches previous results for general and directed acyclic graphs with simpler algorithms.

Provides improved approximation schemes for planar, undirected, and dense graphs.

Abstract

The Restricted Shortest Path (RSP) problem, also known as the Delay-Constrained Least-Cost (DCLC) problem, is an NP-hard bicriteria optimization problem on graphs with $n$ vertices and $m$ edges. In a graph where each edge is assigned a cost and a delay, the goal is to find a min-cost path which does not exceed a delay bound. In this paper, we present improved approximation schemes for RSP on several graph classes. For planar graphs, undirected graphs with positive integer resource (= delay) values, and graphs with $m \in \Omega(n \log n)$, we obtain $(1 + \varepsilon)$-approximations in time $O(mn/\varepsilon)$. For general graphs and directed acyclic graphs, we match the results by Xue et al. (2008, [10]) and Ergun et al. (2002, [1]), respectively, but with arguably simpler algorithms.