具有有限回溯预算的恶意在线二分匹配的紧下界
A tight lower bound for malicious online bipartite matching with limited recourse budget
AI总结：
本文针对带回溯的在线二分匹配问题，证明恶意匹配设定的Ω(n log² n)下界，否定了其存在O(n log n)上界的猜想，确定了该问题恶意变体的渐近回溯复杂度，并补充了扩张器图的O(n log n)上界。
AI中文摘要：
我们研究带回溯的单侧在线二分匹配问题。该设定中，二分图的一侧顶点预先已知，另一侧顶点随时间在线到达并附带其关联边；每次顶点到达后，算法需维持一个最大基数匹配，同时最小化总重分配次数，该次数也被称为回溯预算。尽管已有大量相关研究，该问题的确切回溯复杂度仍未明确：最佳下界为Ω(n log n)，而最佳上界为O(n log² n)，其中n表示在线顶点的数量。仅在森林等受限图类中，已知存在O(n log n)的紧上界。已知最佳上界由一个非常简单且自然的算法SAP（最短增广路径算法）实现，该算法在每次顶点到达后应用最短增广路径，且人们推测该算法是最优的。对该算法的所有已知上界分析均不依赖于算法维持的特定最大匹配，因此这些分析也适用于一个更困难的问题，我们称之为恶意匹配设定：每次顶点到达后，维持的匹配会被替换为下一步的最坏情况最大匹配。这引发了一个猜想，即恶意设定仍能达到O(n log n)的回溯界，与原始模型的推测最优复杂度一致。我们的主要结果是恶意匹配设定的Ω(n log² n)下界，从而否定了该猜想。结合之前的上界，这确定了该问题恶意变体的渐近回溯复杂度。我们还为扩张器图补充了O(n log n)的上界。
英文摘要：
We study one-sided online bipartite matching with recourse. In this setting, one side of a bipartite graph is known in advance, while vertices on the other side arrive online together with their incident edges. After each arrival, the algorithm must maintain a maximum-cardinality matching while minimizing the total number of reallocations, also known as the recourse budget. Despite extensive work, the exact recourse complexity of the problem remains unsettled: the best lower bound is $Ω(n \log n)$, whereas the best upper bound is $\mathcal{O}(n \log^2 n)$, where $n$ denotes the number of online vertices. Tight upper bounds of $\mathcal{O}(n \log n)$ are known only for restricted graph classes, such as forests. The best known upper bounds are attained by a very simple and natural algorithm SAP, which after each arrival applies a shortest augmenting path, and it is conjectured to be optimal. All known upper bound analyses of this algorithm do not depend on the particular maximum matching maintained by the algorithm. Consequently, they also apply to a more difficult problem, which we call the malicious matching setting: after each arrival, the maintained matching is replaced by a worst-case maximum matching for the next step. This led to the conjecture that the malicious setting still admits an $\mathcal{O}(n \log n)$ recourse bound, in line with the conjectured optimal complexity of the original model. Our main result is an $Ω(n \log^2 n)$ lower bound for the malicious matching setting, thus disproving the conjecture. Together with the previous upper bound, this settles the asymptotic recourse complexity of the malicious variant of the problem. We complement our lower bound with an upper bound of $\mathcal{O}(n \log n)$ for expander graphs.
