短增广路径与受限顶点重分配下的确定性在线匹配

中文摘要 AI 辅助

我们研究具有局部改动的确定性在线二分匹配问题。在线顶点逐一到达，并揭示与固定离线集合之间的边。每次到达后，算法维护一个匹配，并且只能通过从新顶点开始的长度为1或3的增广路径来更新匹配。在预算模型OMP_3(s,t)中，每个离线顶点最多可被重新分配s次，每个在线顶点最多可被重新分配t次。我们的主要结果是确定性算法最低成本路径(LCP)，该算法在每种预算选择下都能达到最优竞争比。若s=0或t=0，最佳可能比为1/2。对于所有s≥1和有限t，最优比为γ_t=(2·2^t-1)/(3·2^t-1)。该值在t=1时等于3/5，并随着t增加收敛至2/3。对于t=∞，最优比恰好为2/3。特别地，允许每个离线顶点一次重新分配就已经达到任何更大离线预算下可达到的最佳保证。我们还证明了所有确定性算法的匹配上界。我们的分析是原始-对偶的，并使用LCP历史图的结构描述，该描述解释了耗尽的重新分配预算如何阻止长度为3的增广。我们还研究了离线加权变体。特别地，我们确定了两个单位离线预算情况下的精确最优确定性比：WOMP_3(1,1)为2-√2，WOMP_3(1,∞)为(√5-1)/2。两者都严格低于其未加权对应值。如果将权重放在在线顶点上，则没有确定性算法具有正竞争比。

英文摘要

We study deterministic online bipartite matching with local recourse. Online vertices arrive one by one and reveal edges to a fixed offline set. After each arrival, the algorithm maintains a matching and may update it only by an augmenting path of length $1$ or $3$ starting at the new vertex. In the budgeted model $\operatorname{OMP}_3(s,t)$, each offline vertex can be reassigned at most $s$ times and each online vertex at most $t$ times. Our main result is a deterministic algorithm, Lowest-Cost-Path ($\operatorname{LCP}$), that achieves the optimal competitive ratio for every choice of budgets. If $s=0$ or $t=0$, the best possible ratio is $1/2$. For every $s\ge 1$ and finite $t$, the optimal ratio is $$ γ_t=\frac{2\cdot 2^t-1}{3\cdot 2^t-1}. $$ This value equals $3/5$ for $t=1$ and converges to $2/3$ as $t$ increases. For $t=\infty$, the optimal ratio is exactly $2/3$. In particular, allowing a single reassignment per offline vertex already matches the best guarantee achievable with any larger offline budget. We also prove matching upper bounds for all deterministic algorithms. Our analysis is primal--dual and uses a structural description of the history graph of $\operatorname{LCP}$, which explains how exhausted reassignment budgets can block length-$3$ augmentations. We also study offline-weighted variants. In particular, we determine the exact optimal deterministic ratios for two unit-offline-budget cases: $2-\sqrt2$ for $\operatorname{WOMP}_3(1,1)$, and $(\sqrt5-1)/2$ for $\operatorname{WOMP}_3(1,\infty)$. Both are strictly below their unweighted counterparts. If weights are placed on online vertices instead, no deterministic algorithm has a positive competitive ratio.

发表机构

• Department of Mathematical Engineering, Universidad de Chile（智利大学数学工程系）
• Center for Mathematical Modeling CNRS-IRL 2807, Universidad de Chile（智利大学数学建模中心）

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