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

We develop polynomial-size LP-relaxations for orienteering and the 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 regret at most a given bound R that cover all nodes, where the regret of an r-v path is its length − crv. For rooted orienteering, we introduce a natural bidirected LP-relaxation and obtain a simple 3approximation algorithm via LP-rounding. This is the first LP-based guarantee for this problem. We also show that point-to-point (P2P) 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 [14]. 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.