Fixed-Parameter Tractability and Completeness I: Basic Results

We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism which has running time 2 O(b) N O(1) , where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe fpt algorithms for certain permutation group problems that are used as subroutines in our algorithm. Keywords and phrases IntroductionA hypergraph is an ordered pair X = (V, E) where V is the vertex set and E ⊆ 2 V is the edge set. Two hypergraphs X = (V, E) and X = (V , E ) are said to be isomorphic,Given two hypergraphs X and X , the decision problem Hypergraph Isomorphism (HI) asks whether X ∼ = X . Graph Isomorphism (GI) is obviously polynomial-time reducible to HI. Conversely, HI is also known to be polynomial-time reducible to GI: Given a pair of hypergraphs X = (V, E) and X = (V , E ) as instance for HI, the reduced instance of GI consists of two corresponding bipartite graphs Y and Y defined as follows. The graph Y has vertex set V E and edge set E(Y ) = {{v, e} | v ∈ V, e ∈ E and v ∈ e}, and Y is defined similarly. Here, C D denotes the disjoint union of the sets C and D. It is easy to verify that Y ∼ = Y if and only if X ∼ = X assuming that V can be mapped only to V and E can be mapped only to E . This latter condition is easy to enforce. However, since the above reduction blows up the size of the vertex set in the bipartite encoding, the Zemlyachenko-Luks-Babai graph isomorphism algorithm [3,5,6,25] that runs in time c √ n log n , where n is the size of the vertex set of the graph, does not yield a similar algorithm for hypergraph isomorphism. We note here that the best known hypergraph isomorphism test due to Luks [16] has running time c n . Recently, Babai and Codenotti [4] have shown a 2Õ (k 2 √ n) isomorphism testing algorithm for hypergraphs with hyperedges of size bounded by k.Motivated by this situation, we explore the same algorithmic problem for bounded color class hypergraphs. The input to Colored Hypergraph Isomorphism (CHI) is a pair

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“…Pioneered by Downey and Fellows in [8], it is a flourishing area of research (see, e.g. the monographs [9,11]).…”

Section: Parametrized Complexity and Isomorphism Testingmentioning

confidence: 99%

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

et al. 2013

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“…Perhaps the classification would provide a hierarchy of structure classes, parameterized by some measure k, and a fixed-parameter tractability result for this classification is possible, as in the work of Downey et al [10].…”

Section: Basics On Rna Secondary Structurementioning

confidence: 99%

Problems on RNA Secondary Structure Prediction and Design

Condon

2003

Automata, Languages and Programming

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“…However, one difference between the two, perhaps no more than a historical accident, is that NP was originally defined in terms of resource bounds on a machine model, and the discovery that it has complete problems under polynomial-time reductions (and indeed that many natural combinatorial problems are NP-complete) came as a major advance, which also shows the robustness of the class. On the other hand, the classes W [t] were originally defined as the sets of problems reducible to certain natural complete problems by means of fixed-parameter tractable (fpt) reductions [5]. These classes therefore have complete problems by construction.…”

Section: Introductionmentioning

confidence: 99%

Parameterized Complexity Classes under Logical Reductions

Dawar

2009

Mathematical Foundations of Computer Science 2009

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Abstract. The parameterized complexity classes of the W -hierarchy are usually defined as the problems reducible to certain natural complete problems by means of fixed-parameter tractable (fpt) reductions. We investigate whether the classes can be characterised by means of weaker, logical reductions. We show that each class W [t] has complete problems under slicewise bounded-variable firstorder reductions. These are a natural weakening of slicewise bounded-variable LFP reductions which, by a result of Flum and Grohe, are known to be equivalent to fpt-reductions. If we relax the restriction on having a bounded number of variables, we obtain reductions that are too strong and, on the other hand, if we consider slicewise quantifier-free first-order reductions, they are considerably weaker. These last two results are established by considering the characterisation of W [t] as the closure of a class of Fagin-definability problems under fpt-reductions. We show that replacing these by slicewise first-order reductions yields a hierarchy that collapses, while allowing only quantifier-free first-order reductions yields a hierarchy that is provably strict.

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Fixed-Parameter Tractability and Completeness I: Basic Results

Cited by 335 publications

References 66 publications

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

Problems on RNA Secondary Structure Prediction and Design

Parameterized Complexity Classes under Logical Reductions

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