We study the robust communication complexity of maximum matching. Edges of an arbitrary $n$-vertex graph $G$ are randomly partitioned between Alice and Bob independently and uniformly. Alice has to send a single message to Bob such that Bob can find an (approximate) maximum matching of the whole graph $G$. We specifically study the best approximation ratio achievable via protocols where Alice communicates only $\widetilde{O}(n)$ bits to Bob. There has been a growing interest on the robust communication model due to its connections to the random-order streaming model. An algorithm of Assadi and Behnezhad [ICALP'21] implies a $(2/3+\epsilon_0 \sim .667)$-approximation for a small constant $0 < \epsilon_0 < 10^{-18}$, which remains the best-known approximation for general graphs. For bipartite graphs, Assadi and Behnezhad [Random'21] improved the approximation to .716 albeit with a computationally inefficient (i.e., exponential time) protocol. In this paper, we study a natural and efficient protocol implied by a random-order streaming algorithm of Bernstein [ICALP'20] which is based on edge-degree constrained subgraphs (EDCS) [Bernstein and Stein; ICALP'15]. The result of Bernstein immediately implies that this protocol achieves an (almost) $(2/3 \sim .666)$-approximation in the robust communication model. We present a new analysis, proving that it achieves a much better (almost) $(5/6 \sim .833)$-approximation. This significantly improves previous approximations both for general and bipartite graphs. We also prove that our analysis of Bernstein's protocol is tight.

翻译：我们研究最大匹配的鲁棒通信复杂度。任意$n$顶点图$G$的边被独立均匀地随机划分给Alice和Bob。Alice需向Bob发送一条消息，使Bob能找出整个图$G$的（近似）最大匹配。我们特别关注在Alice仅向Bob传输$\widetilde{O}(n)$比特的协议中可达的最佳近似比。由于鲁棒通信模型与随机顺序流模型的联系，对该模型的研究日益增长。Assadi和Behnezhad [ICALP'21] 的算法给出了一个$(2/3+\epsilon_0 \sim .667)$-近似比，其中$0 < \epsilon_0 < 10^{-18}$为小常数，这仍是针对一般图已知的最佳近似比。对于二分图，Assadi和Behnezhad [Random'21] 将近似比改进至.716，但该协议计算效率低下（即指数时间）。在本文中，我们研究由Bernstein [ICALP'20] 的随机顺序流算法所蕴含的一个自然且高效的协议，该算法基于边度约束子图（EDCS）[Bernstein and Stein; ICALP'15]。Bernstein的结果直接表明，该协议在鲁棒通信模型中实现了（几乎）$(2/3 \sim .666)$-近似比。我们提出一种新的分析方法，证明该协议能达到更好的（几乎）$(5/6 \sim .833)$-近似比。这显著改进了针对一般图和二分图的先前近似结果。我们还证明了我们对Bernstein协议的分析是紧的。