39 research outputs found

    Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth

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    We give a fixed-parameter tractable algorithm that, given a parameter kk and two graphs G1,G2G_1,G_2, either concludes that one of these graphs has treewidth at least kk, or determines whether G1G_1 and G2G_2 are isomorphic. The running time of the algorithm on an nn-vertex graph is 2O(k5logk)n52^{O(k^5\log k)}\cdot n^5, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in 2O(k5logk)n52^{O(k^5\log k)}\cdot n^5 time that, for a given graph GG on nn vertices, either concludes that the treewidth of GG is at least kk, or: * finds in an isomorphic-invariant way a graph c(G)\mathfrak{c}(G) that is isomorphic to GG; * finds an isomorphism-invariant construction term --- an algebraic expression that encodes GG together with a tree decomposition of GG of width O(k4)O(k^4). Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for G1G_1 and G2G_2 are equal.Comment: Full version of a paper presented at FOCS 201

    Mine 'Em All: A Note on Mining All Graphs

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    International audienceWe study the complexity of the problem of enumerating all graphs with frequency at least 1 and computing their support. We show that there are hereditary classes of graphs for which the complexity of this problem depends on the order in which the graphs should be enumerated (e.g. from frequent to infrequent or from small to large). For instance, the problem can be solved with polynomial delay for databases of planar graphs when the enumerated graphs should be output from large to small but it cannot be solved even in incremental-polynomial time when the enumerated graphs should be output from most frequent to least frequent (unless P=NP)

    Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs

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    The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width are known to be solvable in polynomial time [Bod90],[YBFT99]. We give restricted space algorithms for these problems proving the following results: - Isomorphism for bounded tree distance width graphs is in L and thus complete for the class. We also show that for this kind of graphs a canon can be computed within logspace. - For bounded treewidth graphs, when both input graphs are given together with a tree decomposition, the problem of whether there is an isomorphism which respects the decompositions (i.e. considering only isomorphisms mapping bags in one decomposition blockwise onto bags in the other decomposition) is in L. - For bounded treewidth graphs, when one of the input graphs is given with a tree decomposition the isomorphism problem is in LogCFL. - As a corollary the isomorphism problem for bounded treewidth graphs is in LogCFL. This improves the known TC1 upper bound for the problem given by Grohe and Verbitsky [GroVer06].Comment: STACS conference 2010, 12 page

    Graph Isomorphism in Quasipolynomial Time Parameterized by Treewidth

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    We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and develop a quasipolynomial-time algorithm for the multiple-coset isomorphism problem. The algorithm for the multiple-coset isomorphism problem allows to exploit graph decompositions of the given input graphs within Babai's group-theoretic framework. We use it to develop a graph isomorphism test that runs in time npolylog(k)n^{\operatorname{polylog}(k)} where nn is the number of vertices and kk is the minimum treewidth of the given graphs and polylog(k)\operatorname{polylog}(k) is some polynomial in log(k)\operatorname{log}(k). Our result generalizes Babai's quasipolynomial-time graph isomorphism test.Comment: 52 pages, 1 figur

    Canonizing Graphs of Bounded Tree Width in Logspace

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    Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized by logarithmic-space (logspace) algorithms. This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous class of graphs of bounded tree width one of the few classes of graphs for which the complexity of the isomorphism problem has been exactly determined.Comment: 26 page

    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

    Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs

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    We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph HH as a minor to graphs excluding HH as a topological subgraph. We prove that for a fixed HH, every graph excluding HH as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, we prove that such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter H|H|. We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a "typical" application of the structure theorem, we show that on graphs excluding HH as a topological subgraph, Partial Dominating Set (find kk vertices whose closed neighborhood has maximum size) can be solved in time f(H,k)nO(1)f(H,k)\cdot n^{O(1)} time. More significantly, we show that on graphs excluding HH as a topological subgraph, Graph Isomorphism can be solved in time nf(H)n^{f(H)}. This result unifies and generalizes two previously known important polynomial-time solvable cases of Graph Isomorphism: bounded-degree graphs and HH-minor free graphs. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition
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