Parameterized Complexity of Small Weight Automorphisms and Isomorphisms

Abstract

We study the parameterized complexity of computing nontrivial automorphisms of weight k for a given hypergraph $X=(V,E)$, with k as fixed parameter, where the weight of a permutation $\pi \in S_n$ is the number of points moved by $\pi$. Building on the earlier work of Schweitzer (in: Proceedings of 19th ESA, Springer, Berlin, 2011. https://doi.org/10.1007/978-3-642-23719-5_32), we show the following results: (1) Computing nontrivial automorphisms of weight at most k for d-hypergraphs (that is, with edge-size bounded by d) remains fixed parameter tractable, with d treated as a second fixed parameter. Likewise, finding isomorphisms of weight k between d-hypergraphs X and Y (both defined on vertex set [n]) remains fixed parameter tractable. (2) For dealing with the exact weight k version of the problem, we introduce a more general algorithmic problem PermCode: given a permutation group G by a generating set and a fixed parameter k, is there is a nontrivial element of G with support at most (or exactly) k? We give a method for shrinking large orbits of the given group G to obtain subgroups while maintaining existence of weight at most k elements in it. An application of this yields an FPT algorithm for finding exact weight k nontrivial automorphisms in d-hypergraphs, d as second fixed parameter. (3) For hypergraphs with edges of unbounded size, we show that the problem is in $\textsf {FPT } ^{\textsc {GI}}$. (4) Computing d-hypergraph isomorphisms of weight exactly k is fixed parameter tractable. This requires a more complicated orbit shrinking technique.