Approximation and Inapproximability of Non-Preemptive Dial-a-Ride

Abstract

The non-preemptive dial-a-ride problem (DaRP) asks for a minimum-cost route of a capacitated vehicle that serves a collection of pickup-and-delivery requests, without allowing objects to be transferred or temporarily unloaded. We study this problem and the closely related minimum-ratio Steiner $k$-Forest (S$k$F) in general symmetric metrics on $n$ locations. We develop a randomized polynomial-time algorithm with expected approximation ratio $\mathcal{O}(n^{1/4}\log^{11}n)$ for the minimum-ratio S$k$F. Our algorithm combines a deterministic approximation on hierarchically well-separated trees with randomized metric embeddings. Using this algorithm, we obtain expected $\mathcal{O}(n^{1/4}\log^{12}n)$-approximations for both the non-preemptive DaRP and the S$k$F. These guarantees improve on the previous $n^{0.3+\varepsilon}$-approximation ratios for the S$k$F and the non-preemptive DaRP, where $\varepsilon>0$ is any fixed constant. Complementing these algorithmic results, we establish the first superconstant inapproximability bounds for the non-preemptive DaRP. Under the Exponential Time Hypothesis (ETH), no deterministic polynomial-time algorithm achieves an $x^{1/(\log\log x)^c}$-approximation for some constant $c>0$, where $x$ denotes either the number of locations $n$ or the vehicle capacity $k$. Under the quantitative Gap-ETH, we rule out every deterministic polynomial-time $x^{o(1)}$-approximation. These bounds hold even on unit-edge stars with $n=(2+o(1))k$.