Solving maximum clique in sparse graphs: an $$O(nm+n2^{d/4})$$ O ( n m + n 2 d / 4 ) algorithm for $$d$$ d -degenerate graphs

Buchanan, Austin; Walteros, Jose L.; Butenko, Sergiy; Pardalos, Pãnos M.

doi:10.1007/s11590-013-0698-2

Cited by 21 publications

(18 citation statements)

References 8 publications

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“…A power law graph is a graph in which the number of vertices with degree d is proportional to x α where 1 ≤ α ≤ 3. When 1 < α ≤ 2 we have f = O(n 1/2α ), and when 2 < α < 3 we have f = O(n (3−α)/4 ) [42]. Combining with the running time, O(f n3 f /3 ) of the Bron-Kerbosch variant [40], we find that the running time for finding all maximal cliques in a power law graph to be 2 O( √ n) .…”

Section: Complexitymentioning

confidence: 84%

The network you keep: Analyzing persons of interest using cliqster

Fadaee

Farajtabar

Sundaram

et al. 2014

2014 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM 2014)

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Our goal is to determine the structural differences between different categories of networks and to use these differences to predict the network category. Existing work on this topic has looked at social networks such as Facebook, Twitter, co-author networks etc. We, instead, focus on a novel data set that we have assembled from a variety of sources, including law-enforcement agencies, financial institutions, commercial database providers and other similar organizations. The data set comprises networks of persons of interest with each network belonging to different categories such as suspected terrorists, convicted individuals etc. We demonstrate that such "anti-social" networks are qualitatively different from the usual social networks and that new techniques are required to identify and learn features of such networks for the purposes of prediction and classification.We propose Cliqster, a new generative Bernoulli process-based model for unweighted networks. The generating probabilities are the result of a decomposition which reflects a network's community structure. Using a maximum likelihood solution for the network inference leads to a least-squares problem. By solving this problem, we are able to present an efficient algorithm for transforming the network to a new space which is both concise and discriminative. This new space preserves the identity of the network as much as possible. Our algorithm is interpretable and intuitive. Finally, by comparing our research against the baseline method (SVD) and against a state-of-the-art Graphlet algorithm, we show the strength of our algorithm in discriminating between different categories of networks.

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Section: Complexitymentioning

confidence: 84%

The network you keep: Analyzing persons of interest using cliqster

Fadaee

Farajtabar

Sundaram

et al. 2014

2014 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM 2014)

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“…The paper [11] presents the correction procedure of only a single computer error and in the papers [12,13] the group of authors has suggested MCP solution for a degenerate graph and the approaches to finding the maximum quasicliques. So, it seems promising to develop the method of determining the maximum cliques in non-oriented graphs that makes possible the execution of different applications in information systems in a real-time scale.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…In graph G / , vertices (7,8,9,12) relate to all the vertices in G / , so delete them with their incident ribs. In this case, bring them into set U={7, 8, 9, 12}, and graph G / will be as shown in Fig.…”

Section: Fig 4 Graph G /mentioning

confidence: 99%

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Development of maximum clique definition method in a non-oriented graph

Listrovoy

Butenko

Bryksin

et al. 2017

EEJET

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“…However, there is a simple, linear‐time heuristic that is provably best. It relies on a greedy, degree‐based ordering, which is common in exact algorithms for the maximum clique problem .…”

Section: Some Provably Best Heuristicsmentioning

confidence: 99%

On provably best construction heuristics for hard combinatorial optimization problems

et al. 2015

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In this article, a heuristic is said to be provably best if, assuming P ≠ N P , no other heuristic always finds a better solution (when one exists). This extends the usual notion of “best possible” approximation algorithms to include a larger class of heuristics. We illustrate the idea on several problems that are somewhat stylized versions of real‐life network optimization problems, including the maximum clique, maximum k‐club, minimum (connected) dominating set, and minimum vertex coloring problems. The corresponding provably best construction heuristics resemble those commonly used within popular metaheuristics. Along the way, we show that it is hard to recognize whether the clique number and the k‐club number of a graph are equal, yet a polynomial‐time computable function is “sandwiched” between them. This is similar to the celebrated Lovász function wherein an efficiently computable function lies between two graph invariants that are N P ‐hard to compute. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(3), 238–245 2016

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Solving maximum clique in sparse graphs: an $$O(nm+n2^{d/4})$$ O ( n m + n 2 d / 4 ) algorithm for $$d$$ d -degenerate graphs

Cited by 21 publications

References 8 publications

The network you keep: Analyzing persons of interest using cliqster

The network you keep: Analyzing persons of interest using cliqster

Development of maximum clique definition method in a non-oriented graph

On provably best construction heuristics for hard combinatorial optimization problems

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