In this paper we consider the multiparty equality problem in graphs, where every vertex of a graph $G$ is given an input, and the goal of the vertices is to decide whether all inputs are equal. We study this problem in the local broadcast model, where a message sent by a vertex is received by all its neighbors and the total cost of a protocol is the sum of the lengths of the messages sent by the vertices. This setting was studied by Khan and Vaidya, who gave in 2021 a protocol achieving a 4-approximation in the general case. We study this multiparty communication problem through the lens of network topology. We design a new protocol for 2-connected graphs, whose efficiency relies on the notion of total vertex cover in graph theory. This protocol outperforms the aforementioned 4-approximation in a number of cases. To demonstrate its applicability, we apply it to obtain optimal or asymptotically optimal protocols for several natural network topologies such as cycles, hypercubes, and grids. On the way we also provide new bounds of independent interest on the size of total vertex covers in regular graphs.

翻译：本文研究图上的多方相等性问题，其中图$G$的每个顶点被赋予一个输入，顶点的目标是判定所有输入是否相等。我们在局部广播模型中研究该问题，其中顶点发送的消息会被其所有邻居接收，协议的总成本为各顶点发送消息的长度之和。Khan与Vaidya于2021年针对该设置提出了一种在一般情况下实现4-近似比的协议。我们通过网络拓扑的视角研究这一多方通信问题，为2-连通图设计了一种新协议，其效率依赖于图论中的全顶点覆盖概念。该协议在多种情况下优于前述的4-近似方案。为展示其适用性，我们将其应用于环、超立方体、网格等自然网络拓扑，获得了最优或渐近最优的协议。在此过程中，我们还针对正则图的全顶点覆盖规模提出了具有独立意义的新界。