Distributed Testing of Excluded Subgraphs

TL;DR

This paper explores distributed property testing for H-freeness in graphs within the CONGEST model, showing that testing for small forbidden subgraphs like triangles and 4-node graphs is efficient, but larger ones are inherently harder.

Contribution

It proves that testing for K_4 and C_4-freeness can be done in constant rounds, while larger K_k and C_k are significantly more challenging, highlighting the limits of generic testing algorithms.

Findings

Constant-round testing for K_4 and C_4-freeness is possible.

Testing larger K_k and C_k-freeness requires more than constant rounds.

DFS and BFS testers fail for k>4, indicating inherent hardness.

Abstract

We study property testing in the context of distributed computing, under the classical CONGEST model. It is known that testing whether a graph is triangle-free can be done in a constant number of rounds, where the constant depends on how far the input graph is from being triangle-free. We show that, for every connected 4-node graph H, testing whether a graph is H-free can be done in a constant number of rounds too. The constant also depends on how far the input graph is from being H-free, and the dependence is identical to the one in the case of testing triangles. Hence, in particular, testing whether a graph is K_4-free, and testing whether a graph is C_4-free can be done in a constant number of rounds (where K_k denotes the k-node clique, and C_k denotes the k-node cycle). On the other hand, we show that testing K_k-freeness and C_k-freeness for k>4 appear to be much harder. Specifically, we investigate two natural types of generic algorithms for testing H-freeness, called DFS tester and BFS tester. The latter captures the previously known algorithm to test the presence of triangles, while the former captures our generic algorithm to test the presence of a 4-node graph pattern H. We prove that both DFS and BFS testers fail to test K_k-freeness and C_k-freeness in a constant number of rounds for k>4.