A Tight (1.5+?)-Approximation for Unsplittable Capacitated Vehicle Routing on Trees

In the unsplittable capacitated vehicle routing problem (UCVRP) on trees, we are given a rooted tree with edge weights and a subset of vertices of the tree called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the root of the tree such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1. For the special case when all terminals have equal demands, a long line of research culminated in a quasi-polynomial time approximation scheme [Jayaprakash and Salavatipour, TALG 2023] and a polynomial time approximation scheme [Mathieu and Zhou, TALG 2023]. In this work, we study the general case when the terminals have arbitrary demands. Our main contribution is a polynomial time (1.5+?)-approximation algorithm for the UCVRP on trees. This is the first improvement upon the 2-approximation algorithm more than 30 years ago. Our approximation ratio is essentially best possible, since it is NP-hard to approximate the UCVRP on trees to better than a 1.5 factor

Robust minimum energy wireless routing for underwater acoustic communication networks

Marine robots are an increasingly attractive means for observing and monitoring the ocean, but underwater acoustic communications remain a major challenge. The channel exhibits long delay spreads with frequency-dependent attenuation; moreover, it is time-varying. We consider the minimum energy wireless transmission problem [MET], augmented by the practical condition that constraints on link power must be satisfied in probability. For this, we formulate the robust counterpart of the multicommodity mixed-integer linear programming (MILP) model from Haugland and Yuan [1], and derive scaled power levels that account for uncertainty. Our main result is that the deterministic formulation with these scaled power levels recovers exactly the optimal robust solution in the absence of correlations, and therefore allows for efficient solution via MILP. This approach achieves significant power improvements over heuristics, and naturally lends itself to vehicle networks.United States. Office of Naval Research (Grant N00014-09-1-0700