Constant-Factor Approximation to Deadline TSP and Related Problems in (Almost) Quasi-Polytime

Abstract

We investigate a genre of vehicle-routing problems (VRPs), that we call max-reward VRPs, wherein nodes located in a metric space have associated rewards that depend on their visiting times, and we seek a path that earns maximum reward. A prominent problem in this genre is deadline TSP, where nodes have deadlines and we seek a path that visits all nodes by their deadlines and earns maximum reward. Our main result is a constant-factor approximation for deadline TSP running in time O(n^O(log(nΔ))) in metric spaces with integer distances at most Δ. This is the first improvement over the approximation factor of O(log n) due to Bansal et al. [N. Bansal et al., 2004] in over 15 years (but is achieved in super-polynomial time). Our result provides the first concrete indication that log n is unlikely to be a real inapproximability barrier for deadline TSP, and raises the exciting possibility that deadline TSP might admit a polytime constant-factor approximation. 
At a high level, we obtain our result by carefully guessing an appropriate sequence of O(log (nΔ)) nodes appearing on the optimal path, and finding suitable paths between any two consecutive guessed nodes. We argue that the problem of finding a path between two consecutive guessed nodes can be relaxed to an instance of a special case of deadline TSP called point-to-point (P2P) orienteering. Any approximation algorithm for P2P orienteering can then be utilized in conjunction with either a greedy approach, or an LP-rounding approach, to find a good set of paths overall between every pair of guessed nodes. While concatenating these paths does not immediately yield a feasible solution, we argue that it can be covered by a constant number of feasible solutions. Overall our result therefore provides a novel reduction showing that any α-approximation for P2P orienteering can be leveraged to obtain an O(α)-approximation for deadline TSP in O(n^O(log nΔ)) time.
Our results extend to yield the same guarantees (in approximation ratio and running time) for a substantial generalization of deadline TSP, where the reward obtained by a client is given by an arbitrary non-increasing function (specified by a value oracle) of its visiting time. Finally, we discuss applications of our results to variants of deadline TSP, including settings where both end-nodes are specified, nodes have release dates, and orienteering with time windows.