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.
(1) The query complexity of $O(1/\log n)$-testing asymmetry of $n$-vertex graphs is ${\widetilde\Omega}({\sqrt{n/\log n}})$, even if the tested graph is guaranteed to consist of connected components of size $O(\log n)$.
(2) For $s(n)=\Omega(\log n)$, the query complexity of $\e$-testing the set of asymmetric $n$-vertex graphs in which each connected component has size at most $s(n)$ is at most $O({\sqrt n}\cdot s(n)/\e)$ and at least $\Omega({\sqrt{n^{1-O(\e)}/s(n)}})$.
In addition, we show that testing asymmetry in the dense graph model is almost trivial.
Reframing and restructuring the results.
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 $\epsilon$-testing asymmetry (in this case) is at most $O({\sqrt n}\cdot s(n)/\epsilon)$
and at least $\Omega({\sqrt{n^{1-O(\epsilon)}/s(n)}})$. 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.
Clarifying the proof of Prop 7.
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 $\epsilon$-testing asymmetry (in this case) is at most $O({\sqrt n}\cdot s(n)/\epsilon)$ and at least $\Omega({\sqrt{n^{1-O(\epsilon)}/s(n)}})$. 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.
Correcting inaccuracies introduced in yesterday's revision.
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.
The result was generalized to offer lower bounds for any value of the proximity parameter. This is supported by the new Proposition 6.
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 $\epsilon$-testing asymmetry (in this case) is at most $O({\sqrt n}\cdot s(n)/\epsilon)$, whereas the query complexity of $o(1/s(n))$-testing asymmetry (in this case) is at least $\Omega({\sqrt{n/s(n)}})$.
In addition, we show that testing asymmetry in the dense graph model is almost trivial.
