Online Hypergraph Matching with Delays

TL;DR

This paper investigates online hypergraph matching with delays, focusing on ridesharing applications, and provides algorithms with provable competitive ratios for utility maximization and cost minimization, considering various vehicle capacities and delay constraints.

Contribution

It introduces new algorithms and bounds for online hypergraph matching with delays, including optimal competitive ratios for utility maximization and cost minimization, applicable to different vehicle capacities.

Findings

Optimal competitive ratio for utility maximization is 1/d for k ≥ 3.

Polynomial-time thresholding algorithm achieves 3/2 ratio for k=2 in cost minimization.

Randomized batching algorithm is (2 - 1/d) log k-competitive for k>2, with NP-hardness bounds.

Abstract

We study an online hypergraph matching problem with delays, motivated by ridesharing applications. In this model, users enter a marketplace sequentially, and are willing to wait up to $d$ timesteps to be matched, after which they will leave the system in favor of an outside option. A platform can match groups of up to $k$ users together, indicating that they will share a ride. Each group of users yields a match value depending on how compatible they are with one another. As an example, in ridesharing, $k$ is the capacity of the service vehicles, and $d$ is the amount of time a user is willing to wait for a driver to be matched to them. We present results for both the utility maximization and cost minimization variants of the problem. In the utility maximization setting, the optimal competitive ratio is $\frac{1}{d}$ whenever $k \geq 3$, and is achievable in polynomial-time for any fixed $k$. In the cost minimization variation, when $k = 2$, the optimal competitive ratio for deterministic algorithms is $\frac{3}{2}$ and is achieved by a polynomial-time thresholding algorithm. When $k>2$, we show that a polynomial-time randomized batching algorithm is $(2 - \frac{1}{d}) \log k$-competitive, and it is NP-hard to achieve a competitive ratio better than $\log k - O (\log \log k)$.