We initiate a thorough study of distributed property testing -producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called dense graph testing model we emulate sequential tests for nearly all graph properties having 1-sided tests, while in the general and sparse models we obtain faster tests for trianglefreeness, cycle-freeness and bipartiteness, respectively. In addition, we show a logarithmic lower bound for testing bipartiteness and cycle-freeness, which holds even in the stronger LOCAL model.In most cases, aided by parallelism, the distributed algorithms have a much shorter running time as compared to their counterparts from the sequential querying model of traditional property testing. The simplest property testing algorithms allow a relatively smooth transitioning to the distributed model. For the more complex tasks we develop new machinery that may be of independent interest.

We show here that every non-adaptive property testing algorithm making a constant number of queries, over a fixed alphabet, can be converted to a sample-based (as per [Goldreich and Ron, 2015]) testing algorithm whose average number of queries is a fixed, smaller than 1, power of n. Since the query distribution of the sample-based algorithm is not dependent at all on the property, or the original algorithm, this has many implications in scenarios where there are many properties that need to be tested for concurrently, such as testing (relatively large) unions of properties, or converting a Merlin-Arthur Proximity proof (as per [Gur and Rothblum, 2013]) to a proper testing algorithm.The proof method involves preparing the original testing algorithm for a combinatorial analysis, which in turn involves a new result about the existence of combinatorial structures (essentially generalized sunflowers) that allow the sample-based tester to replace the original constant query complexity tester.

Abstract. We study optimization versions of Graph Isomorphism. Given two graphs G1, G2, we are interested in finding a bijection π from V (G1) to V (G2) that maximizes the number of matches (edges mapped to edges or non-edges mapped to non-edges). We give an n O(log n) time approximation scheme that for any constant factor α < 1, computes an α-approximation. We prove this by combining the n O(log n) time additive error approximation algorithm of Arora et al. [Math. Program., 92, 2002] with a simple averaging algorithm. We also consider the corresponding minimization problem (of mismatches) and prove that it is NP-hard to α-approximate for any constant factor α. Further, we show that it is also NP-hard to approximate the maximum number of edges mapped to edges beyond a factor of 0.94. We also explore these optimization problems for bounded color class graphs which is a well studied tractable special case of Graph Isomorphism. Surprisingly, the bounded color class case turns out to be harder than the uncolored case in the approximate setting.

Let G = S be a solvable permutation group of the symmetric group S n given as input by the generating set S. We give a deterministic polynomial-time algorithm that computes an expanding generating set of size O(n 2 ) for G. More precisely, the algorithm computes a subset T ⊂ G of size O(n 2 )(1/λ) O(1) such that the undirected Cayley graph Cay(G, T ) is a λ-spectral expander (the O notation suppresses log O(1) n factors). As a byproduct of our proof, we get a new explicit construction of ε-bias spaces of size O(n poly(log d))( 1 ε ) O(1) for the groups Z n d . The earlier known size bound was O((d + n/ε 2 )) 11/2 given by [AMN98].

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