论文标题
在线超图与延迟匹配
Online Hypergraph Matching with Delays
论文作者
论文摘要
我们研究了一个在线超图匹配问题，该问题与乘车应用程序有关。在此模型中，用户依次进入市场，并愿意等待$ d $ timesteps匹配，之后他们将使系统不支持外部选项。一个平台可以将高达$ k $的用户组成的组匹配，表明他们将共享乘车。每组用户都会产生匹配值，这取决于他们彼此之间的兼容性。例如，在乘车共享中，$ k $是服务车辆的容量，而$ d $是用户愿意等待驾驶员与他们匹配的时间。 我们介绍了问题的实用性最大化和成本最小化变体的结果。在实用程序最大化设置中，只要$ k \ geq 3 $，最佳竞争比为$ \ frac {1} {d} $，并且对于任何固定的$ k $，都可以在多项式时间内实现。在成本最小化的变化中，当$ k = 2 $时，确定性算法的最佳竞争比为$ \ frac {3} {2} $，并且是通过多项式时间阈值算法实现的。当$ k> 2 $时，我们表明多项式随机批处理算法为$（2 - \ frac {1} {d}）\ log k $ copetive，并且获得比$ \ log k-o（\ log \ log \ log \ log \ log \ k）$更好的竞争比是NP-HARD。
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 $\log k - O (\log \log k)$.
