On The Multiparty Communication Complexity of Testing Triangle-Freeness

In this paper we initiate the study of property testing in simultaneous and non-simultaneous multi-party communication complexity, focusing on testing triangle-freeness in graphs. We consider the $\textit{coordinator}$ model, where we have $k$ players receiving private inputs, and a coordinator who receives no input; the coordinator can communicate with all the players, but the players cannot communicate with each other. In this model, we ask: if an input graph is divided between the players, with each player receiving some of the edges, how many bits do the players and the coordinator need to exchange to determine if the graph is triangle-free, or $\textit{far}$ from triangle-free? For general communication protocols, we show that $\tilde{O}(k(nd)^{1/4}+k^2)$ bits are sufficient to test triangle-freeness in graphs of size $n$ with average degree $d$ (the degree need not be known in advance). For $\textit{simultaneous}$ protocols, where there is only one communication round, we give a protocol that uses $\tilde{O}(k \sqrt{n})$ bits when $d = O(\sqrt{n})$ and $\tilde{O}(k (nd)^{1/3})$ when $d = Ω(\sqrt{n})$; here, again, the average degree $d$ does not need to be known in advance. We show that for average degree $d = O(1)$, our simultaneous protocol is asymptotically optimal up to logarithmic factors. For higher degrees, we are not able to give lower bounds on testing triangle-freeness, but we give evidence that the problem is hard by showing that finding an edge that participates in a triangle is hard, even when promised that at least a constant fraction of the edges must be removed in order to make the graph triangle-free.