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

TL;DR

This paper introduces compact LP relaxations and LP-rounding algorithms for orienteering and regret-bounded vehicle routing, achieving new approximation guarantees and improving computational efficiency.

Contribution

It presents the first LP-based approximation for rooted orienteering and new LP formulations that improve approximation ratios for regret-bounded vehicle routing.

Findings

A 3-approximation algorithm for rooted orienteering via LP-rounding.

Reduction of point-to-point orienteering to regret-version with factor-2 loss.

A 15-approximation algorithm for regret-bounded vehicle routing using new LPs.

Abstract

We develop polynomial-size LP-relaxations for {\em orienteering} and the {\em regret-bounded vehicle routing problem} (\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 \rvrp, the goal is to find the minimum number of $r$-rooted paths of {\em 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 {\em rooted orienteering}, we introduce a natural bidirected LP-relaxation and obtain a simple $3$-approximation algorithm via LP-rounding. This is the {\em first LP-based} guarantee for this problem. We also show that {\em point-to-point} (\ptp) {\em orienteering} can be reduced to a regret-version of rooted orienteering at the expense of a factor-2 loss in approximation. For \rvrp, we propose two compact LPs that lead to significant improvements, in both approximation ratio and running time, over the approach in~\cite{FriggstadS14}. One of these is a natural modification of the LP for rooted orienteering; the other is an unconventional formulation that is motivated by certain structural properties of an \rvrp-solution, which leads to a $15$-approximation algorithm for \rvrp.