On the complexity of fixed parameter clique and dominating set

Eisenbrand, Friedrich; Grandoni, Fabrizio

doi:10.1016/j.tcs.2004.05.009

Cited by 127 publications

(105 citation statements)

References 13 publications

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“…In general, these are computationally very challenging problems. Given an arbitrary template of size k and a graph with n nodes, the best known rigorous result for the subgraph isomorphism problem is obtained by Eisenbrand et al [10] with a running time of roughly O(n ωk/3 ) (which improves on the naive O(n k ) time), where ω denotes the exponent of the best possible matrix multiplication algorithm. If the template has an independent set of size s, Vassilevska et al [28] give an algorithm with an improved running time of O(2 s n k−s+3 k O (1) ); this is improved slightly by Kowaluk et al [16].…”

Section: Introductionmentioning

confidence: 99%

SAHAD: Subgraph Analysis in Massive Networks Using Hadoop

Zhao

Wang

Butt

et al. 2012

2012 IEEE 26th International Parallel and Distributed Processing Symposium

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Abstract-Relational subgraph analysis, e.g. finding labeled subgraphs in a network, which are isomorphic to a template, is a key problem in many graph related applications. It is computationally challenging for large networks and complex templates. In this paper, we develop SAHAD, an algorithm for relational subgraph analysis using Hadoop, in which the subgraph is in the form of a tree. SAHAD is able to solve a variety of problems closely related with subgraph isomorphism, including counting labeled/unlabeled subgraphs, finding supervised motifs, and computing graphlet frequency distribution. We prove that the worst case work complexity for SAHAD is asymptotically very close to that of the best sequential algorithm. On a mid-size cluster with about 40 compute nodes, SAHAD scales to networks with up to 9 million nodes and a quarter billion edges, and templates with up to 12 nodes. To the best of our knowledge, SAHAD is the first such Hadoop based subgraph/subtree analysis algorithm, and performs significantly better than prior approaches for very large graphs and templates. Another unique aspect is that SAHAD is also amenable to running quite easily on Amazon EC2, without needs for any system level optimization.

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Section: Introductionmentioning

confidence: 99%

SAHAD: Subgraph Analysis in Massive Networks Using Hadoop

Zhao

Wang

Butt

et al. 2012

2012 IEEE 26th International Parallel and Distributed Processing Symposium

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show abstract

“…Eisenbrand & Grandoni [9] slightly improve on these bounds for (3k + 2)-CLIQUE (with k ≥ 2) and for (3k + 1)-CLIQUE (with 1 ≤ k ≤ 5). In particular, for 4-CLIQUE [9] gives a time complexity of n 3.334 .…”

Section: Fact the 3k-clique Problem For A P-vertex Graph Can Be Solvmentioning

confidence: 88%

Space and Time Complexity of Exact Algorithms: Some Open Problems

Woeginger

2004

Lecture Notes in Computer Science

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“…Instead, we want to use matrix multiplication. We define three functions a 12 , a 13 , a 23 : V (G) × V (G) → N which we later interpret as matrices whose indices range over…”

Section: V1v2v3 ∈ {0 1} Indicates Whether the Mapping β(T) → {Vmentioning

confidence: 99%

Homomorphisms are a good basis for counting small subgraphs

Curticapean

Dell

Marx

2017

Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

182

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We introduce graph motif parameters, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs H, the number of H-copies (induced or not) in an input graph G, and the number of homomorphisms from H to G.Using the framework of graph motif parameters, we obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G: For graphs H on k edges, we show how to count subgraph copies ofby a surprisingly simple algorithm. This improves upon previously known running times, such as O(n 0.91k+c ) time for k-edge matchings or O(n 0.46k+c ) time for k-cycles. Furthermore, we prove a general complexity dichotomy for evaluating graph motif parameters: Given a class C of such parameters, we consider the problem of evaluating f ∈ C on input graphs G, parameterized by the number of induced subgraphs that f depends upon. For every recursively enumerable class C, we prove the above problem to be either FPT or #W[1]-hard, with an explicit dichotomy criterion. This allows us to recover known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms in a uniform and simplified way, together with improved lower bounds.Finally, we extend graph motif parameters to colored subgraphs and prove a complexity trichotomy:

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On the complexity of fixed parameter clique and dominating set

Cited by 127 publications

References 13 publications

SAHAD: Subgraph Analysis in Massive Networks Using Hadoop

SAHAD: Subgraph Analysis in Massive Networks Using Hadoop

Space and Time Complexity of Exact Algorithms: Some Open Problems

Homomorphisms are a good basis for counting small subgraphs

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