Abstract:
For a permutation group $G$ acting on the set $\Omega$ we say that two strings $x,y\,:\,\Omega\to\boole$ are {\em $G$-isomorphic} if they are equivalent under the action of $G$, \ie, if for some $\pi\in G$ we have $x(i^{\pi})=y(i)$ for all $i\in\Omega$. Cyclic Shift, Graph Isomorphism and Hypergraph Isomorphism are special cases, and subcases corresponding to certain classes of groups have been central to the design of efficient isomorphism testing for subclasses of graphs (Luks 1982). We study the complexity of $G$-isomorphism in the context of property testing: we want to find the randomized decision tree complexity of distinguishing the cases when $x$ and $y$ are $G$-isomorphic from the cases when they are at least $\delta$-far from being $G$-isomorphic (in normalized Hamming distance). Error can be 1-sided or 2-sided. In each case we consider two models. In the query-1 model we assume $y$ is known and only $x$ needs to be queried. In the query-2 model we have to query both $x$ and $y$. We give various upper and lower bounds for the four combinations of models considered in terms of $n=|\Omega|$ and $|G|$. In many cases, substantial gaps remain between the upper and lower bounds. However, for {\em primitive permutation groups}, we obtain a tight (up to polylog($n$) factors) bound of $\wti{\Theta}(\sqrt{n\log |G|})$ for the 1-sided error query complexity in the query-2 model and a tight bound of $\wti{\Theta}(\log |G|)$ for the 1-sided error query complexity in the query-1 model. This result extends results of Fischer and Matsliah (2006) on Graph Isomorphism to a surprisingly general class of groups which also includes isomorphism of uniform hypergraphs of any rank. Besides the fact that they include Graph Isomorphism, primitive permutation groups are significant because they form the ``building blocks'' of all permutations groups, providing the base cases of a natural divide-and-conquer approach successfully exploited in algorithm design (Luks, 1982). While all our bounds are in terms of the order of $G$, it seems likely that tighter bounds will depend on the finer structure of $G$; our result on primitive groups is a first step in this direction.