Distributed Property Testing for Subgraph-Freeness Revisited

In the subgraph-freeness problem, we are given a constant-size graph $H$, and wish to determine whether the network contains $H$ as a subgraph or not. The \emph{property-testing} relaxation of the problem only requires us to distinguish graphs that are $H$-free from graphs that are $\epsilon$-far from $H$-free, meaning an $\epsilon$-fraction of their edges must be removed to obtain an $H$-free graph. Recently, Censor-Hillel et. al. and Fraigniaud et al. showed that in the property-testing regime it is possible to test $H$-freeness for any graph $H$ of size 4 in constant time, $O(1/\epsilon^2)$ rounds, regardless of the network size. However, Fraigniaud et. al. also showed that their techniques for graphs $H$ of size 4 cannot test $5$-cycle-freeness in constant time. In this paper we revisit the subgraph-freeness problem and show that $5$-cycle-freeness, and indeed $H$-freeness for many other graphs $H$ comprising more than 4 vertices, can be tested in constant time. We show that $C_k$-freeness can be tested in $O(1/\epsilon)$ rounds for any cycle $C_k$, improving on the running time of $O(1/\epsilon^2)$ of the previous algorithms for triangle-freeness and $C_4$-freeness. In the special case of triangles, we show that triangle-freeness can be solved in $O(1)$ rounds independently of $\epsilon$, when $\epsilon$ is not too small with respect to the number of nodes and edges. We also show that $T$-freeness for any constant-size tree $T$ can be tested in $O(1)$ rounds, even without the property-testing relaxation. Building on these results, we define a general class of graphs for which we can test subgraph-freeness in $O(1/\epsilon)$ rounds. This class includes all graphs over 5 vertices except the 5-clique, $K_5$. For cliques $K_s$ over $s \geq 3$ nodes, we show that $K_s$-freeness can be tested in $O(m^{1/2-1/(s-2)}/\epsilon^{1/2+1/(s-2)})$ rounds, where $m$ is the number of edges.