使用Expander分解的在线拼车
Online Carpooling using Expander Decompositions
摘要：
我们考虑在线拼车问题：给定$n$顶点，一系列边随着时间的推移到达。当边$e_t = (u_t, v_t)$到达时间步长$t$时，算法必须将边定向为$v_t \rightarrow u_t$或$u_t \rightarrow v_t$，目的是最小化任何顶点的最大差异，即，它的入度和出度之间的绝对差。边缘对应于想要一起乘坐的成对的人，并且定向表示指定驾驶员。然后，差异目标对应于每个人驾驶接近他们参与的乘坐的公平份额。在本文中，我们设计了有效的算法，可以保持polylog $(n,T)$最大差异（w.h.p）的任何序列的$T$到达时，到达的边缘是独立和均匀地从任何给定的图$G$采样。这为在线（随机）拼车问题提供了第一个多对数边界。在此之前的工作，最有名的界限是$O(\sqrt{n \log n})$ -差异的任何敌对序列的到来，或$O(\log\!\log n)$ -差异界限的随机到来时，$G$是完整的图形。我们的论文的技术关键是在表明，简单的贪婪算法，它具有可证明的好的差异界限时，到达的边缘均匀随机从完整的图，也有polylog差异时，$G$是一个扩展图。然后，我们将其与已知的扩展器分解结果相结合，设计我们的整体算法。
英文摘要：
We consider the online carpooling problem: given $n$ vertices, a sequence of edges arrive over time. When an edge $e_t = (u_t, v_t)$ arrives at time step $t$, the algorithm must orient the edge either as $v_t \rightarrow u_t$ or $u_t \rightarrow v_t$, with the objective of minimizing the maximum discrepancy of any vertex, i.e., the absolute difference between its in-degree and out-degree. Edges correspond to pairs of persons wanting to ride together, and orienting denotes designating the driver. The discrepancy objective then corresponds to every person driving close to their fair share of rides they participate in. In this paper, we design efficient algorithms which can maintain polylog$(n,T)$ maximum discrepancy (w.h.p) over any sequence of $T$ arrivals, when the arriving edges are sampled independently and uniformly from any given graph $G$. This provides the first polylogarithmic bounds for the online (stochastic) carpooling problem. Prior to this work, the best known bounds were $O(\sqrt{n \log n})$-discrepancy for any adversarial sequence of arrivals, or $O(\log\!\log n)$-discrepancy bounds for the stochastic arrivals when $G$ is the complete graph. The technical crux of our paper is in showing that the simple greedy algorithm, which has provably good discrepancy bounds when the arriving edges are drawn uniformly at random from the complete graph, also has polylog discrepancy when $G$ is an expander graph. We then combine this with known expander-decomposition results to design our overall algorithm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
