Compact, Provably-Good LPs for Orienteering and Regret-Bounded Vehicle Routing

Abstract

We develop polynomial-size LP-relaxations for orienteering and the regret-bounded vehicle routing problem ($\mathsf {RVRP}$) and devise suitable LP-rounding algorithms that lead to various new insights and approximation results for these problems. In orienteering, the goal is to find a maximum-reward r-rooted path, possibly ending at a specified node, of length at most some given budget B. In $\mathsf {RVRP}$, the goal is to find the minimum number of r-rooted paths of regret at most a given bound R that cover all nodes, where the regret of an r-v path is its length − $c_{rv}$. For rooted orienteering, we introduce a natural bidirected LP-relaxation and obtain a simple 3-approximation algorithm via LP-rounding. This is the first LP-based guarantee for this problem. We also show that point-to-point ($\mathsf {P2P}$) orienteering can be reduced to a regret-version of rooted orienteering at the expense of a factor-2 loss in approximation. For $\mathsf {RVRP}$, we propose two compact LPs that lead to significant improvements, in both approximation ratio and running time, over the approach in [10]. One is a natural modification of the LP for rooted orienteering; the other is an unconventional formulation motivated by certain structural properties of an $\mathsf {RVRP}$-solution, which leads to a 15-approximation for $\mathsf {RVRP}$.

A full version of the paper is available on the CS arXiv.

Z. Friggstad—Research supported by Canada Research Chairs program and NSERC Discovery Grant.

C. Swamy—Research supported by NSERC grant 327620-09 and an NSERC DAS Award.