We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts. The common denominator of these results is a deterministic distributed rounding method for certain linear programs , which is the first such rounding method, to our knowledge. A sampling of our end results is as follows: An $$O\mathopen {}\left( \log ^2 \Delta \cdot \log n\right) \mathclose {}$$ O log 2 Δ · log n -round deterministic distributed algorithm for computing a maximal matching, in n -node graphs with maximum degree $$\Delta $$ Δ . This is the first improvement in about 20 years over the celebrated $$O(\log ^4 n)$$ O ( log 4 n ) -round algorithm of Hańćkowiak, Karoński, and Panconesi [SODA’98, PODC’99]. A deterministic distributed algorithm for computing a $$(2+\varepsilon )$$ ( 2 + ε ) -approximation of maximum matching in $$O\mathopen {}\left( \log ^2 \Delta \cdot \log \frac{1}{\varepsilon } + \log ^ * n\right) \mathclose {}$$ O log 2 Δ · log 1 ε + log ∗ n rounds. This is exponentially faster than the classic $$O(\Delta +\log ^* n)$$ O ( Δ + log ∗ n ) -round 2-approximation of Panconesi and Rizzi [DIST’01]. With some modifications, the algorithm can also find an almost maximal matching which leaves only an $$\varepsilon $$ ε -fraction of the edges on unmatched nodes. An $$O\mathopen {}\left( \log ^2 \Delta \cdot \log \frac{1}{\varepsilon } \cdot \log _{1+\varepsilon } W + \log ^ * n\right) \mathclose {}$$ O log 2 Δ · log 1 ε · log 1 + ε W + log ∗ n -round deterministic distributed algorithm for computing a $$(2+\varepsilon )$$ ( 2 + ε ) -approximation of a maximum weighted matching, and also for the more general problem of maximum weighted b -matching. Here, W denotes the maximum normalized weight. These improve over the $$O\mathopen {}\left( \log ^4 n \cdot \log _{1+\varepsilon } W\right) \mathclose {}$$ O log 4 n · log 1 + ε W -round $$(6+\varepsilon )$$ ( 6 + ε ) -approximation algorithm of Panconesi and Sozio [DIST’10].

中文翻译：

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通过舍入改进确定性分布式匹配

我们针对一些经过充分研究的匹配问题提出了改进的确定性分布式算法，这些算法比已知的对应算法更简单、更快、更准确和/或更通用。这些结果的共同点是某些线性程序的确定性分布式舍入方法，据我们所知，这是第一个这样的舍入方法。我们的最终结果示例如下： An $$O\mathopen {}\left( \log ^2 \Delta \cdot \log n\right) \mathclose {}$$ O log 2 Δ · log n -round用于计算最大匹配的确定性分布式算法，在具有最大度数 $$\Delta $$ Δ 的 n 节点图中。这是对 Hańćkowiak、Karoński 和 Panconesi [SODA'98, PODC'99] 著名的 $$O(\log ^4 n)$O ( log 4 n ) -round 算法大约 20 年来的第一次改进。用于计算 $$(2+\varepsilon )$$ ( 2 + ε ) - $$O\mathopen {}\left( \log ^2 \Delta \cdot \log \frac) 中最大匹配的近似的确定性分布式算法{1}{\varepsilon } + \log ^ * n\right) \mathclose {}$$ O log 2 Δ · log 1 ε + log ∗ n 轮。这比 Panconesi 和 Rizzi [DIST'01] 的经典 $$O(\Delta +\log ^* n)$$ O ( Δ + log ∗ n ) -round 2-approximation 快得多。通过一些修改，该算法还可以找到几乎最大的匹配，它只留下未匹配节点上的边的 $$\varepsilon $$ ε -分数。一个 $$O\mathopen {}\left( \log ^2 \Delta \cdot \log \frac{1}{\varepsilon } \cdot \log _{1+\varepsilon } W + \log ^ * n\right ) \mathclose {}$$ O log 2 Δ · log 1 ε · log 1 + ε W + log ∗ n -round 确定性分布式算法，用于计算 $$(2+\varepsilon )$$ ( 2 + ε ) - 近似最大加权匹配，以及最大加权 b 匹配的更一般问题。这里，W 表示最大归一化权重。这些改进了 $$O\mathopen {}\left( \log ^4 n \cdot \log _{1+\varepsilon } W\right) \mathclose {}$$ O log 4 n · log 1 + ε W -round $$(6+\varepsilon )$$ ( 6 + ε ) - Panconesi 和 Sozio [DIST'10] 的近似算法。