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 \[ \Omega(\frac{1}{\log\log{n}}) \leq \beta \leq 1-\Theta(\frac{\log^*{n}}{{\log{n}}}) \] 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$(1-\epsilon)$-近似最大二分匹配的半流式算法需要\[ \Omega(\frac{\log{(1/\epsilon)}}{{\log{(1/\beta)}}}) \]遍，其中$\beta \in (0,1)$是使得存在包含$n^{\beta}$个边不相交且大小为$\Theta(n)$的诱导匹配的$n$顶点图（此类图称为RS图）的最大参数。目前已知\[ \Omega(\frac{1}{\log\log{n}}) \leq \beta \leq 1-\Theta(\frac{\log^*{n}}{{\log{n}}}) \]，而缩小这一上下界之间的巨大差距一直是组合学中著名的难题。在$\beta = \Omega(1)$这一合理假设下，我们的下界结果首次为半流式模型中匹配的（小）常数近似提供了遍数-近似下界，这是图流式文献中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。此前相关工作通过将此类问题归约到布尔域，并借助XOR引理和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用拼接替代先前的XOR引理，提出针对排列的困难性放大结果。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分析排列得到证明，且可能具有独立的研究价值。