On the benefits of adaptivity in property testing of dense graphs

We consider the question of whether adaptivity can improve the complexity of property testing algorithms in the dense graphs model. It is known that there can be at most a quadratic gap between adaptive and non-adaptive testers in this model, but it was not known whether any gap indeed exists. In this work we reveal such a gap. Specifically, we focus on the well studied property of bipartiteness. Bogdanov and Trevisan (IEEE Symposium on Computational Complexity, 2004) proved a lower bound of Ω(1/ε²) on the query complexity of non-adaptive testing algorithms for bipartiteness. This lower bound holds for graphs with maximum degree O(εn). Our main result is an adaptive testing algorithm for bipartiteness of graphs with maximum degree O(en) whose query complexity is¹ Õ(1/ε^3/2). A slightly modified version of our algorithm can be used to test the combined property of being bipartite and having maximum degree O(εn). Thus we demonstrate that adaptive testers are stronger than non-adaptive testers in the dense graphs model. We note that the upper bound we obtain is tight up-to polylogarithmic factors, in view of the Ω(1/ε^3/2) lower bound of Bogdanov and Trevisan for adaptive testers. In addition we show that Õ(1/ε^3/2) queries also suffice when (almost) all vertices have degree Ω(√ε · n). In this case adaptivity is not necessary.