Parameterized Complexity of Small Weight Automorphisms and Isomorphisms

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 π∈ S_n is the number of points moved by π . 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 ^GI . (4) Computing d -hypergraph isomorphisms of weight exactly k is fixed parameter tractable. This requires a more complicated orbit shrinking technique.