On Testing Asymmetry in the Bounded Degree Graph Model

Webpage for a paper by Oded Goldreich


We consider the problem of testing asymmetry in the bounded-degree graph model, where a graph is called asymmetric if the identity permutation is its only automorphism. Seeking to determine the query complexity of this testing problem, we provide partial results. Considering the special case of $n$-vertex graphs with connected components of size at most $s(n)=\Omega(\log n)$, we show that the query complexity of $\e$-testing asymmetry (in this case) is at most $O({\sqrt n}\cdot s(n)/\e)$ and at least $\Omega({\sqrt{n^{1-O(\e)}/s(n)}})$. In particular, the query complexity of $o(1/s(n))$-testing asymmetry is at least $\Omega({\sqrt{n/s(n)}})$. In addition, we show that testing asymmetry in the dense graph model is almost trivial.

Note: The versions of 2020 present a weaker result. Specifically, in January 2021 the result was generalized to offer lower bounds for any value of the proximity parameter. This is supported by the new Proposition 6.

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