Approximation Schemes for the Restricted Shortest Path Problem

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 ∈ Ω(n log n), we obtain (1 + ε)-approximations in time O (mn/ε). 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.