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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 $log k - O (log log k)$.
In this paper, we study $k$-Way Min-cost Perfect Matching with Delays - the $k$-MPMD problem. This problem considers a metric space with $n$ nodes. Requests arrive at these nodes in an online fashion. The task is to match these requests into sets of
For online matching with the line metric, we present a lower bound of $Omega(log n)$ on the approximation ratio of any online (possibly randomized) algorithm. This beats the previous best lower bound of $Omega(sqrt{log n})$ and matches the known upper bound of $O(log n)$.
Online bipartite matching with edge arrivals remained a major open question for a long time until a recent negative result by [Gamlath et al. FOCS 2019], who showed that no online policy is better than the straightforward greedy algorithm, i.e., no o
We study the problem of matching agents who arrive at a marketplace over time and leave after d time periods. Agents can only be matched while they are present in the marketplace. Each pair of agents can yield a different match value, and the planner
We study the greedy-based online algorithm for edge-weighted matching with (one-sided) vertex arrivals in bipartite graphs, and edge arrivals in general graphs. This algorithm was first studied more than a decade ago by Korula and Pal for the biparti