151 research outputs found

    Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable

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    Reduction Techniques for Graph Isomorphism in the Context of Width Parameters

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    We study the parameterized complexity of the graph isomorphism problem when parameterized by width parameters related to tree decompositions. We apply the following technique to obtain fixed-parameter tractability for such parameters. We first compute an isomorphism invariant set of potential bags for a decomposition and then apply a restricted version of the Weisfeiler-Lehman algorithm to solve isomorphism. With this we show fixed-parameter tractability for several parameters and provide a unified explanation for various isomorphism results concerned with parameters related to tree decompositions. As a possibly first step towards intractability results for parameterized graph isomorphism we develop an fpt Turing-reduction from strong tree width to the a priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure

    Parameterized Complexity of Small Weight Automorphisms

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    We show that checking if a given hypergraph has an automorphism that moves exactly k vertices is fixed parameter tractable, using k and additionally either the maximum hyperedge size or the maximum color class size as parameters. In particular, it suffices to use k as parameter if the hyperedge size is at most polylogarithmic in the size of the given hypergraph. As a building block for our algorithms, we generalize Schweitzer\u27s FPT algorithm [ESA 2011] that, given two graphs on the same vertex set and a parameter k, decides whether there is an isomorphism between the two graphs that moves at most k vertices. We extend this result to hypergraphs, using the maximum hyperedge size as a second parameter. Another key component of our algorithm is an orbit-shrinking technique that preserves permutations that move few points and that may be of independent interest. Applying it to a suitable subgroup of the automorphism group allows us to switch from bounded hyperedge size to bounded color classes in the exactly-k case

    An Improved Isomorphism Test for Bounded-Tree-Width Graphs

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    We give a new fpt algorithm testing isomorphism of n-vertex graphs of tree width k in time 2^{k polylog(k)} poly n, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time 2^{O(k^5 log k)}poly n. Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree width k. Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai\u27s algorithm as a black box in one place. We give a second algorithm which, at the price of a slightly worse run time 2^{O(k^2 log k)}poly n, avoids the use of Babai\u27s algorithm and, more importantly, has the additional benefit that it can also be used as a canonization algorithm

    Isomorphism Testing for Graphs Excluding Small Minors

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    We prove that there is a graph isomorphism test running in time npolylog(h)n^{\operatorname{polylog}(h)} on nn-vertex graphs excluding some hh-vertex graph as a minor. Previously known bounds were npoly(h)n^{\operatorname{poly}(h)} (Ponomarenko, 1988) and npolylog(n)n^{\operatorname{polylog}(n)} (Babai, STOC 2016). For the algorithm we combine recent advances in the group-theoretic graph isomorphism machinery with new graph-theoretic arguments

    Hypergraph Isomorphism for Groups with Restricted Composition Factors

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    We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group ? over domain V, and asking whether there is a permutation ? ? ? that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time (n+m)^O((log d)^c) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for the above problem due to Schweitzer and Wiebking (STOC 2019) runs in time n^O(d)m^O(1). As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K_{3,h} as a minor in time n^O((log h)^c). In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time n^O((log g)^c)
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