AI 中文总结
本研究针对边权重无感知匹配问题，提出Harmonic Ranking算法，其竞争比达0.698，优于此前相关工作，且在顶点权重随机到达模型中也有改进的在线竞争比，结果经计算机辅助验证。
AI 中文摘要
我们研究边权重无感知二分匹配问题：每条潜在边的权重已知，但仅在探测时才会揭示其存在，且两个自由顶点间的成功探测必须立即接受。我们给出了一个明确的随机算法，其竞争比为0.698，改进了此前Huang、Sun、Wu和Zhao（FOCS 2025）给出的最佳保证0.659。该结果由计算机辅助，并通过可复现的精确整数计算验证。同一算法在顶点权重随机到达模型中具有0.698竞争比的在线实现，改进了Mahdian和Yan（STOC 2011）的0.696无权重保证，以及Peng和Tang（EC 2025）的0.686顶点权重保证。我们的算法Harmonic Ranking是\textsc{Ranking}的角色对称泛化，它为每个顶点分配独立随机秩$x_z$，并按$w_{uv}\ rac{h(x_u)h(x_v)}{h(x_u)+h(x_v)}$的递减顺序探测潜在边$uv$。该调和优先级源于预算平衡的收益分割和相互提议解释。分析将两条截断曲线提升为指示函数，将指数规模的因子揭示问题简化为多项式规模的有向最小割实例。容量向下取整的最大流计算给出了严格证明。此外，我们观察到因子揭示程序的有限网格无权重松弛与Mahdian-Yan程序完全一致。
英文摘要
We study edge-weighted oblivious bipartite matching. The weight of every potential edge is known, but its existence is revealed only when the edge is probed, and a successful probe between two free vertices must be accepted immediately. We give an explicit randomized algorithm with certified competitive ratio $0.698$, improving the previous best guarantee of $0.659$ (Huang, Sun, Wu, and Zhao, FOCS 2025). The result is computer-assisted and verified by a reproducible exact-integer computation. The same algorithm has a $0.698$-competitive online implementation for the vertex-weighted random-arrival model, improving the previous $0.696$ unweighted guarantee of Mahdian and Yan (STOC 2011) and the $0.686$ vertex-weighted guarantee of Peng and Tang (EC 2025). Our algorithm, Harmonic Ranking, is a role-symmetric generalization of \textsc{Ranking}. It assigns an independent random rank $x_z$ to each vertex and probes a potential edge $uv$ in decreasing order of \[ w_{uv}\frac{h(x_u)h(x_v)}{h(x_u)+h(x_v)}. \] This harmonic priority arises from a budget-balanced gain split and a mutual-proposal interpretation. The analysis lifts two cutoff curves into indicators, reducing the exponential-size factor-revealing problem to a polynomial-size directed minimum-cut instance. A maximum-flow computation with rounded-down integer capacities gives a rigorous certificate. Independently, we observe that the finite-grid unweighted relaxation of our factor-revealing program coincides exactly with a Mahdian--Yan program.
Comments44 pages, 4 figures
