凸二分图上的在线匹配
Online Matching in Convex Bipartite Graphs
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中文总结 AI 辅助
本文研究凸二分图上的在线匹配问题，针对均匀长度模型提出Flip随机算法，证明其竞争比为2/3且为紧值，同时给出该模型下随机算法竞争比的上界3/4。
中文摘要 AI 辅助
在线资源分配系统（如门诊排班和频谱分配）常需将陆续到达的请求分配给有序的稀缺资源池，每个请求接受连续区间内的可行选项。我们研究凸二分图上的在线匹配问题，该问题需满足决策不可撤销且请求为对抗性到达。首先，我们证明仅凸性无法改进经典的最坏情况保证1-1/e，该保证由Ranking算法实现。接着，我们考虑均匀长度模型，其中每个在线请求恰好有d个连续的离线邻居。我们提出Flip算法，该算法使用一个随机位预先承诺选择最早可行分配或最晚可行分配。尽管这两种自然确定性策略都会浪费容量，且渐近竞争比仅为1/2，但我们证明它们的随机混合策略的竞争比为2/3。该保证对Flip算法是紧的，且对半自适应对抗者依然有效，半自适应对抗者会在选择到达顺序前观察所选策略。我们还证明，在均匀长度模型中，没有任何随机在线算法能达到严格大于3/4的竞争比。
英文摘要
Online resource-allocation systems, like outpatient scheduling and spectrum allocation, often assign sequentially arriving requests to an ordered pool of scarce resources, where each request accepts a contiguous interval of feasible options. We study the resulting online matching problem on convex bipartite graphs under irrevocable decisions and adversarial arrivals. We first show that convexity alone does not improve the classic worst-case guarantee of 1-1/e, achieved by Ranking. We then consider the uniform-length model, in which every online request has exactly d consecutive offline neighbors. We propose Flip, which uses one random bit to commit ex-ante to either earliest-feasible assignment or latest-feasible assignment. Although either natural deterministic policy can waste capacity and be asymptotically only 1/2-competitive, we show that their randomized mixture is 2/3-competitive. This guarantee is tight for Flip and remains valid against a semi-adaptive adversary that observes the selected policy before choosing the arrival order. We also prove that no randomized online algorithm can achieve a competitive ratio strictly larger than 3/4 in the uniform-length model.
