We prove that any semi-streaming algorithm for $(1-\epsilon)$-approximation of maximum bipartite matching requires \[ \Omega(\frac{\log{(1/\epsilon)}}{{\log{(1/\beta)}}}) \] passes, where $\beta \in (0,1)$ is the largest parameter so that an $n$-vertex graph with $n^{\beta}$ edge-disjoint induced matchings of size $\Theta(n)$ exist (such graphs are referred to as RS graphs). Currently, it is known that \[ \Omega(\frac{1}{\log\log{n}}) \leqslant \beta \leqslant 1-\Theta(\frac{\log^*{n}}{{\log{n}}}) \] and closing this huge gap between upper and lower bounds has remained a notoriously difficult problem in combinatorics. Under the plausible hypothesis that $\beta = \Omega(1)$, our lower bound result provides the first pass-approximation lower bound for (small) constant approximation of matchings in the semi-streaming model, a longstanding open question in the graph streaming literature. Our techniques are based on analyzing communication protocols for compressing (hidden) permutations. Prior work in this context relied on reducing such problems to Boolean domain and analyzing them via tools like XOR Lemmas and Fourier analysis on Boolean hypercube. In contrast, our main technical contribution is a hardness amplification result for permutations through concatenation in place of prior XOR Lemmas. This result is proven by analyzing permutations directly via simple tools from group representation theory combined with detailed information-theoretic arguments, and can be of independent interest.

翻译：我们证明，任何用于最大二分图匹配的 $(1-\epsilon)$-近似半流算法都需要 \[ \Omega(\frac{\log{(1/\epsilon)}}{{\log{(1/\beta)}}}) \] 遍（pass），其中 $\beta \in (0,1)$ 是使得存在 $n$ 个顶点、包含 $n^{\beta}$ 条边不相交诱导匹配（各匹配大小为 $\Theta(n)$）的图（这类图称为 RS 图）的最大参数。目前已知 \[ \Omega(\frac{1}{\log\log{n}}) \leqslant \beta \leqslant 1-\Theta(\frac{\log^*{n}}{{\log{n}}}) \]，且缩小上下界间的巨大差距一直是组合学中公认的难题。在 $\beta = \Omega(1)$ 这一合理假设下，我们的下界结果给出了半流模型中小常数匹配近似问题（小误差常数）的首个遍数-近似下界，这是图流领域长期悬而未决的问题。我们的技术基于分析压缩（隐藏）排列的通信协议。先前工作依赖将此类问题归约到布尔域，并通过 XOR 引理及布尔超立方体上的傅里叶分析等工具进行分析。相比之下，我们的主要技术贡献是通过拼接（concatenation）替代先前的 XOR 引理，实现了排列的难度放大结果。该结果通过群表示论中的简单工具结合详细的信息论论证直接对排列进行分析而得到，可能具有独立的研究价值。