AI 中文总结
研究半流模型下单遍最大匹配问题的最佳近似比率，开发新框架将问题简化为构造蓝图，既能捕捉现有下界，又能设计新蓝图排除更好近似，有望改进该领域开放问题。
AI 中文摘要
在半流模型中，有一个\(n\)顶点图\(G=(V,E)\)，其边以任意顺序在流中到达。目标是对流进行一遍或几遍遍历，使用\(\tilde O(n)\)比特的有限内存，并在最后输出手头问题的解决方案。该领域一个核心的开放问题是通过单遍半流算法确定最大匹配问题可能的最佳近似比率。这个问题有一个简单的\(0.5\)近似算法，通过贪婪地维护最大匹配，尽管经过大量努力，这仍是目前的技术水平。该问题的下界也很少，已知最好的界排除了优于\(1/(1+\ln{(2)}) \sim 0.590\)的近似，使用了受极值图论中RS图文献启发的高度复杂构造。我们开发了一个用于证明半流匹配问题下界的新框架。我们的框架抽象出下界中的极值图论和信息论论证，并将问题简化为构造某些固定大小的图，即蓝图。这些蓝图不仅可以捕捉现有的下界，给出更简单和简洁的论证，而且我们还可以设计新的蓝图，用于排除半流匹配问题的\((8 - 2\sqrt{10})/3 \sim 0.558\)近似。我们相信这种方法本身就很有意义，并能在这个诱人的开放问题上带来进一步改进。
英文摘要
In the semi-streaming model, we have an $n$-vertex graph $G=(V,E)$ whose edges arrive in an arbitrary order in a stream. The goal is to make one or a few passes over the stream, use a limited memory of $\tilde O(n)$ bits, and output a solution to the problem at hand at the end. A central open question in this area is to determine the best approximation ratio possible for the maximum matching problem via single-pass semi-streaming algorithms. This problem admits a simple $0.5$-approximation algorithm, by maintaining a maximal matching greedily, which, despite extensive efforts, has remained the state of the art. Lower bounds for this problem have also been few and far between with best known bounds ruling out better than $1/(1+\ln{(2)}) \sim 0.590$ approximation, using a highly complicated construction motivated by the literature on RS graphs from extremal graph theory. We develop a new framework for proving lower bounds for the semi-streaming matching problem. Our framework abstracts out the extremal graph theory and information theoretic arguments in the lower bounds, and reduces the problem to constructing certain constant-size graphs, which we call blueprints. Not only existing lower bounds can be captured by these blueprints, leading to far simpler and more concise arguments, but also we can design new blueprints that can be used to rule out $(8-2\sqrt{10})/3 \sim 0.558$-approximation for the semi-streaming matching problem. We believe this approach can be of its own independent interest and lead to further improvements on this tantalizing open question.
CommentsFull version of the paper in STOC 2026. 56 pages, 9 figures
