We present a deterministic $(1+\varepsilon)$-approximate maximum matching algorithm in $\mathsf{poly} 1/\varepsilon$ passes in the semi-streaming model, solving the long-standing open problem of breaking the exponential barrier in the dependence on $1/\varepsilon$. Our algorithm exponentially improves on the well-known randomized $(1/\varepsilon)^{O(1/\varepsilon)}$-pass algorithm from the seminal work by McGregor~[APPROX05], the recent deterministic algorithm by Tirodkar with the same pass complexity~[FSTTCS18]. Up to polynomial factors in $1/\varepsilon$, our work matches the state-of-the-art deterministic $(\log n / \log \log n) \cdot (1/\varepsilon)$-pass algorithm by Ahn and Guha~[TOPC18], that is allowed a dependence on the number of nodes $n$. Our result also makes progress on the Open Problem 60 at sublinear.info. Moreover, we design a general framework that simulates our approach for the streaming setting in other models of computation. This framework requires access to an algorithm computing an $O(1)$-approximate maximum matching and an algorithm for processing disjoint $(\mathsf{poly} 1 / \varepsilon)$-size connected components. Instantiating our framework in $\mathsf{CONGEST}$ yields a $\mathsf{poly}(\log{n}, 1/\varepsilon)$ round algorithm for computing $(1+\varepsilon$)-approximate maximum matching. In terms of the dependence on $1/\varepsilon$, this result improves exponentially state-of-the-art result by Lotker, Patt-Shamir, and Pettie~[LPSP15]. Our framework leads to the same quality of improvement in the context of the Massively Parallel Computation model as well.

翻译：我们提出一种在半流式模型中以 $\mathsf{poly} 1/\varepsilon$ 遍历次数运行的确定性 $(1+\varepsilon)$-近似最大匹配算法，解决了长期悬而未决的突破 $1/\varepsilon$ 指数依赖性的难题。该算法在 $1/\varepsilon$ 依赖关系上实现了指数级改进，远超 McGregor 开创性工作 [APPROX05] 中著名的随机化 $(1/\varepsilon)^{O(1/\varepsilon)}$ 遍历算法，以及 Tirodkar 近期提出的具有相同遍历复杂度的确定性算法 [FSTTCS18]。在 $1/\varepsilon$ 的多项式因子范围内，本工作与 Ahn 和 Guha [TOPC18] 提出的当前最优确定性 $(\log n / \log \log n) \cdot (1/\varepsilon)$ 遍历算法（该算法允许依赖节点数 $n$）性能持平。我们的成果也推动了 sublinear.info 上 Open Problem 60 的解决。此外，我们设计了一个通用框架，将其流式处理方法模拟至其他计算模型中。该框架需要两种算法支撑：其一计算 $O(1)$-近似最大匹配，其二处理互不相交的 $(\mathsf{poly} 1 / \varepsilon)$ 规模连通分量。将该框架实例化至 $\mathsf{CONGEST}$ 模型后，可得到计算 $(1+\varepsilon)$-近似最大匹配的 $\mathsf{poly}(\log{n}, 1/\varepsilon)$ 轮算法。在 $1/\varepsilon$ 依赖关系上，此结果较 Lotker、Patt-Shamir 和 Pettie [LPSP15] 的当前最优成果实现指数级改进。该框架同样适用于大规模并行计算模型，并带来同等级别的性能提升。