Incremental maintenance of maximal cliques in a dynamic graph

Das, Apurba; Svendsen, Michael; Tirthapura, Srikanta

doi:10.1007/s00778-019-00540-5

Cited by 32 publications

(18 citation statements)

References 48 publications

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“…k团问题 [22,23] 的定义为: 给定一个无向图, 找到所有包含k个顶点的团. [30] 2010 Cheng [31] 2015 Xu [32] 2017 Sun [33] 2019 Das [34] 2020 Das [35] 2001 Kose [39] 2017 Yu [40] 2019 Yu [75] 1977 Tsukiyama [59] 2004 Makino [60] 2013 Chang [61] 2015 Comin [62] 2016 Conte [63] 1973 Bron [41] 2001 Koch [69] 2004 Tomita [58] 2006 Du [70] 2014 Xu [71] 2019 Li [73] On special graphs On general graphs 1965 Fulkerson [24] 1985 Chiba [27] 1992 Jerrum [28] 1993 Blair [29] 1995 Prisner [25] 2003 Spinrad [26] On uncertain graphs 2016 Mukherjee [36] 2019 Li [37] 图 2 MCE算法的分类 [36,37] . 在动态图上维护更新极大团枚举结果的工作由Stix在2004年首次提出 [30] , 对于图中的一条新增边u → v, Stix对包含u的极大团和包含v的极大团进行笛卡尔积, 生成新图中所有可能存在的极大团并一一验证.…”

Section: 一个图中的极大独立集与其补图中的极大团一一对应因此极大独立集枚举问题是极大团枚举unclassified

Maximal clique enumeration problem on graphs: status and challenges

Xu¹,

Liao²,

Shao³

et al. 2022

Sci. Sin.-Inf.

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In the era of big-data, graph mining has become a popular research topic. The maximal clique enumeration (MCE), as a basic problem in graph theory, has been widely used in different fields. However, considering the complexity of the MCE problem and the rapid growth in the scale of real-world graphs, enumerating the maximal cliques in real-world graphs is time-consuming. A large number of researches have been done to improve the algorithm for MCE problem, or to reduce the execution time by applying various computational optimizations. On the MCE problem, this survey has done the following work: existing researches on the MCE problem are classified; the research status of the MCE problem is introduced in detail; the challenges and future directions of the MCE problem are discussed.

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Section: 一个图中的极大独立集与其补图中的极大团一一对应因此极大独立集枚举问题是极大团枚举unclassified

Maximal clique enumeration problem on graphs: status and challenges

Xu¹,

Liao²,

Shao³

et al. 2022

Sci. Sin.-Inf.

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“…Our approach is to regenerate the query graph whenever a new batch of queries arrives. However, an approach based on a dynamic query graph [9] could improve the overall query processing time, because it would eliminate the need to regenerate the query graph from scratch. -A possible method of reducing the size of the query graph is to represent complex queries which have a pre-determined, regular structure as single nodes in the graph.…”

Section: Limitationsmentioning

confidence: 99%

Making the Most of Parallel Composition in Differential Privacy

Smith

Asghar

Gioiosa

et al. 2021

Proceedings on Privacy Enhancing Technologies

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We show that the ‘optimal’ use of the parallel composition theorem corresponds to finding the size of the largest subset of queries that ‘overlap’ on the data domain, a quantity we call the maximum overlap of the queries. It has previously been shown that a certain instance of this problem, formulated in terms of determining the sensitivity of the queries, is NP-hard, but also that it is possible to use graph-theoretic algorithms, such as finding the maximum clique, to approximate query sensitivity. In this paper, we consider a significant generalization of the aforementioned instance which encompasses both a wider range of differentially private mechanisms and a broader class of queries. We show that for a particular class of predicate queries, determining if they are disjoint can be done in time polynomial in the number of attributes. For this class, we show that the maximum overlap problem remains NP-hard as a function of the number of queries. However, we show that efficient approximate solutions exist by relating maximum overlap to the clique and chromatic numbers of a certain graph determined by the queries. The link to chromatic number allows us to use more efficient approximate algorithms, which cannot be done for the clique number as it may underestimate the privacy budget. Our approach is defined in the general setting of f-differential privacy, which subsumes standard pure differential privacy and Gaussian differential privacy. We prove the parallel composition theorem for f-differential privacy. We evaluate our approach on synthetic and real-world data sets of queries. We show that the approach can scale to large domain sizes (up to 1020000), and that its application can reduce the noise added to query answers by up to 60%.

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“…Use Case 2: Maximal Clique Listing Maximal clique listing, in which one enumerates all maximal cliques (i.e., fully-connected subgraphs not contained in a larger such subgraph) in a graph, is one of core graph mining problems [29,37,42,43,49,67,71,72,79,82,84,95,105,109,111,118,124,125,130]. The recursive backtracking algorithm by Bron and Kerbosch (BK) [24] together with a series of enhancements [42,51,52,117] (see Algorithm 4) is an established and, in practice, the most ecient way of solving this problem.…”

Section: High-performance and Simplicitymentioning

confidence: 99%

GraphMineSuite

et al. 2021

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We propose GraphMineSuite (GMS): the first benchmarking suite for graph mining that facilitates evaluating and constructing high-performance graph mining algorithms. First, GMS comes with a benchmark specification based on extensive literature review, prescribing representative problems, algorithms, and datasets. Second, GMS offers a carefully designed software platform for seamless testing of different fine-grained elements of graph mining algorithms, such as graph representations or algorithm subroutines. The platform includes parallel implementations of more than 40 considered baselines, and it facilitates developing complex and fast mining algorithms. High modularity is possible by harnessing set algebra operations such as set intersection and difference, which enables breaking complex graph mining algorithms into simple building blocks that can be separately experimented with. GMS is supported with a broad concurrency analysis for portability in performance insights, and a novel performance metric to assess the throughput of graph mining algorithms, enabling more insightful evaluation. As use cases, we harness GMS to rapidly redesign and accelerate state-of-the-art baselines of core graph mining problems: degeneracy reordering (by >2X), maximal clique listing (by >9×), k -clique listing (by up to 1.1×), and subgraph isomorphism (by 2.5×), also obtaining better theoretical performance bounds.

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Incremental maintenance of maximal cliques in a dynamic graph

Cited by 32 publications

References 48 publications

Maximal clique enumeration problem on graphs: status and challenges

Maximal clique enumeration problem on graphs: status and challenges

Making the Most of Parallel Composition in Differential Privacy

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