Enumerating Maximum Cliques in Massive Graphs

Cliques refer to subgraphs in an undirected graph such that vertices in each subgraph are pairwise adjacent. The maximum clique problem, to find the clique with most vertices in a given graph, has been extensively studied. Besides its theoretical value as an NP-hard problem, the maximum clique probl...

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Published in:IEEE transactions on knowledge and data engineering Vol. 34; no. 9; pp. 4215 - 4230
Main Authors: Lu, Can, Yu, Jeffrey Xu, Wei, Hao, Zhang, Yikai
Format: Journal Article
Language:English
Published: New York IEEE 01.09.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Anomalies
Apexes
Approximation algorithms
Bioinformatics
Color
community search
Complex networks
Enumeration
Gene expression
Graph theory
Iterative algorithms
Iterative methods
Lower bounds
Maximum clique
Search problems
Searching
Social networking (online)
Social networks
Upper bound
Upper bounds
ISSN:1041-4347, 1558-2191
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Abstract Cliques refer to subgraphs in an undirected graph such that vertices in each subgraph are pairwise adjacent. The maximum clique problem, to find the clique with most vertices in a given graph, has been extensively studied. Besides its theoretical value as an NP-hard problem, the maximum clique problem is known to have direct applications in various fields, such as community search in social networks and social media, team formation in expert networks, gene expression and motif discovery in bioinformatics and anomaly detection in complex networks, revealing the structure and function of networks. However, algorithms designed for the maximum clique problem are expensive to deal with real-world networks. In this paper, we first devise a randomized algorithm for the maximum clique problem. Different from previous algorithms that search from each vertex one after another, our approach RMC , for the randomized maximum clique problem, employs a binary search while maintaining a lower bound <inline-formula><tex-math notation="LaTeX">\underline{\omega _c}</tex-math> <mml:math><mml:munder><mml:msub><mml:mi>ω</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>̲</mml:mo></mml:munder></mml:math><inline-graphic xlink:href="lu-ieq1-3036013.gif"/> </inline-formula> and an upper bound <inline-formula><tex-math notation="LaTeX">\overline{\omega _c}</tex-math> <mml:math><mml:mover><mml:msub><mml:mi>ω</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>¯</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="lu-ieq2-3036013.gif"/> </inline-formula> of <inline-formula><tex-math notation="LaTeX">\omega (G)</tex-math> <mml:math><mml:mrow><mml:mi>ω</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="lu-ieq3-3036013.gif"/> </inline-formula>. In each iteration, RMC attempts to find a <inline-formula><tex-math notation="LaTeX">\omega _t</tex-math> <mml:math><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="lu-ieq4-3036013.gif"/> </inline-formula>-clique where <inline-formula><tex-math notation="LaTeX">\omega _t=\lfloor (\underline{\omega _c}+\overline{\omega _c})/2\rfloor</tex-math> <mml:math><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:munder><mml:msub><mml:mi>ω</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>̲</mml:mo></mml:munder><mml:mo>+</mml:mo><mml:mover><mml:msub><mml:mi>ω</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>¯</mml:mo></mml:mover><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo>⌋</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="lu-ieq5-3036013.gif"/> </inline-formula>. As finding <inline-formula><tex-math notation="LaTeX">\omega _t</tex-math> <mml:math><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="lu-ieq6-3036013.gif"/> </inline-formula> in each iteration is NP-complete, we extract a seed set <inline-formula><tex-math notation="LaTeX">S</tex-math> <mml:math><mml:mi>S</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq7-3036013.gif"/> </inline-formula> such that the problem of finding a <inline-formula><tex-math notation="LaTeX">\omega _t</tex-math> <mml:math><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="lu-ieq8-3036013.gif"/> </inline-formula>-clique in <inline-formula><tex-math notation="LaTeX">G</tex-math> <mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq9-3036013.gif"/> </inline-formula> is equivalent to finding a <inline-formula><tex-math notation="LaTeX">\omega _t</tex-math> <mml:math><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="lu-ieq10-3036013.gif"/> </inline-formula>-clique in <inline-formula><tex-math notation="LaTeX">S</tex-math> <mml:math><mml:mi>S</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq11-3036013.gif"/> </inline-formula> with probability guarantees (<inline-formula><tex-math notation="LaTeX">\geq</tex-math> <mml:math><mml:mo>≥</mml:mo></mml:math><inline-graphic xlink:href="lu-ieq12-3036013.gif"/> </inline-formula><inline-formula><tex-math notation="LaTeX"> 1-n^{-c}</tex-math> <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href="lu-ieq13-3036013.gif"/> </inline-formula>). We propose a novel iterative algorithm to determine the maximum clique by searching a <inline-formula><tex-math notation="LaTeX">k</tex-math> <mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq14-3036013.gif"/> </inline-formula>-clique in <inline-formula><tex-math notation="LaTeX">S</tex-math> <mml:math><mml:mi>S</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq15-3036013.gif"/> </inline-formula> starting from <inline-formula><tex-math notation="LaTeX">k=\underline{\omega _c}+1</tex-math> <mml:math><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:munder><mml:msub><mml:mi>ω</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>̲</mml:mo></mml:munder><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="lu-ieq16-3036013.gif"/> </inline-formula> until <inline-formula><tex-math notation="LaTeX">S</tex-math> <mml:math><mml:mi>S</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq17-3036013.gif"/> </inline-formula> becomes <inline-formula><tex-math notation="LaTeX">\lbrace \rbrace</tex-math> <mml:math><mml:mrow><mml:mo>{</mml:mo><mml:mo>}</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="lu-ieq18-3036013.gif"/> </inline-formula>, when more iterations benefit marginally. Due to the potential inconsistency of maximum clique algorithms, we study the problem of maximum clique enumeration and propose an efficient algorithm RMCE to enumerate all maximum cliques in a given graph. As confirmed by the experiments, both RMC and RMCE are much more efficient and robust than previous solutions, RMC can always find the exact maximum clique, and RMCE can always enumerate all maximum cliques in a given graph.
AbstractList Cliques refer to subgraphs in an undirected graph such that vertices in each subgraph are pairwise adjacent. The maximum clique problem, to find the clique with most vertices in a given graph, has been extensively studied. Besides its theoretical value as an NP-hard problem, the maximum clique problem is known to have direct applications in various fields, such as community search in social networks and social media, team formation in expert networks, gene expression and motif discovery in bioinformatics and anomaly detection in complex networks, revealing the structure and function of networks. However, algorithms designed for the maximum clique problem are expensive to deal with real-world networks. In this paper, we first devise a randomized algorithm for the maximum clique problem. Different from previous algorithms that search from each vertex one after another, our approach RMC , for the randomized maximum clique problem, employs a binary search while maintaining a lower bound [Formula Omitted] and an upper bound [Formula Omitted] of [Formula Omitted]. In each iteration, RMC attempts to find a [Formula Omitted]-clique where [Formula Omitted]. As finding [Formula Omitted] in each iteration is NP-complete, we extract a seed set [Formula Omitted] such that the problem of finding a [Formula Omitted]-clique in [Formula Omitted] is equivalent to finding a [Formula Omitted]-clique in [Formula Omitted] with probability guarantees ([Formula Omitted][Formula Omitted]). We propose a novel iterative algorithm to determine the maximum clique by searching a [Formula Omitted]-clique in [Formula Omitted] starting from [Formula Omitted] until [Formula Omitted] becomes [Formula Omitted], when more iterations benefit marginally. Due to the potential inconsistency of maximum clique algorithms, we study the problem of maximum clique enumeration and propose an efficient algorithm RMCE to enumerate all maximum cliques in a given graph. As confirmed by the experiments, both RMC and RMCE are much more efficient and robust than previous solutions, RMC can always find the exact maximum clique, and RMCE can always enumerate all maximum cliques in a given graph.
Cliques refer to subgraphs in an undirected graph such that vertices in each subgraph are pairwise adjacent. The maximum clique problem, to find the clique with most vertices in a given graph, has been extensively studied. Besides its theoretical value as an NP-hard problem, the maximum clique problem is known to have direct applications in various fields, such as community search in social networks and social media, team formation in expert networks, gene expression and motif discovery in bioinformatics and anomaly detection in complex networks, revealing the structure and function of networks. However, algorithms designed for the maximum clique problem are expensive to deal with real-world networks. In this paper, we first devise a randomized algorithm for the maximum clique problem. Different from previous algorithms that search from each vertex one after another, our approach RMC , for the randomized maximum clique problem, employs a binary search while maintaining a lower bound <inline-formula><tex-math notation="LaTeX">\underline{\omega _c}</tex-math> <mml:math><mml:munder><mml:msub><mml:mi>ω</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>̲</mml:mo></mml:munder></mml:math><inline-graphic xlink:href="lu-ieq1-3036013.gif"/> </inline-formula> and an upper bound <inline-formula><tex-math notation="LaTeX">\overline{\omega _c}</tex-math> <mml:math><mml:mover><mml:msub><mml:mi>ω</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>¯</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="lu-ieq2-3036013.gif"/> </inline-formula> of <inline-formula><tex-math notation="LaTeX">\omega (G)</tex-math> <mml:math><mml:mrow><mml:mi>ω</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="lu-ieq3-3036013.gif"/> </inline-formula>. In each iteration, RMC attempts to find a <inline-formula><tex-math notation="LaTeX">\omega _t</tex-math> <mml:math><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="lu-ieq4-3036013.gif"/> </inline-formula>-clique where <inline-formula><tex-math notation="LaTeX">\omega _t=\lfloor (\underline{\omega _c}+\overline{\omega _c})/2\rfloor</tex-math> <mml:math><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:munder><mml:msub><mml:mi>ω</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>̲</mml:mo></mml:munder><mml:mo>+</mml:mo><mml:mover><mml:msub><mml:mi>ω</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>¯</mml:mo></mml:mover><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo>⌋</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="lu-ieq5-3036013.gif"/> </inline-formula>. As finding <inline-formula><tex-math notation="LaTeX">\omega _t</tex-math> <mml:math><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="lu-ieq6-3036013.gif"/> </inline-formula> in each iteration is NP-complete, we extract a seed set <inline-formula><tex-math notation="LaTeX">S</tex-math> <mml:math><mml:mi>S</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq7-3036013.gif"/> </inline-formula> such that the problem of finding a <inline-formula><tex-math notation="LaTeX">\omega _t</tex-math> <mml:math><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="lu-ieq8-3036013.gif"/> </inline-formula>-clique in <inline-formula><tex-math notation="LaTeX">G</tex-math> <mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq9-3036013.gif"/> </inline-formula> is equivalent to finding a <inline-formula><tex-math notation="LaTeX">\omega _t</tex-math> <mml:math><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="lu-ieq10-3036013.gif"/> </inline-formula>-clique in <inline-formula><tex-math notation="LaTeX">S</tex-math> <mml:math><mml:mi>S</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq11-3036013.gif"/> </inline-formula> with probability guarantees (<inline-formula><tex-math notation="LaTeX">\geq</tex-math> <mml:math><mml:mo>≥</mml:mo></mml:math><inline-graphic xlink:href="lu-ieq12-3036013.gif"/> </inline-formula><inline-formula><tex-math notation="LaTeX"> 1-n^{-c}</tex-math> <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href="lu-ieq13-3036013.gif"/> </inline-formula>). We propose a novel iterative algorithm to determine the maximum clique by searching a <inline-formula><tex-math notation="LaTeX">k</tex-math> <mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq14-3036013.gif"/> </inline-formula>-clique in <inline-formula><tex-math notation="LaTeX">S</tex-math> <mml:math><mml:mi>S</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq15-3036013.gif"/> </inline-formula> starting from <inline-formula><tex-math notation="LaTeX">k=\underline{\omega _c}+1</tex-math> <mml:math><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:munder><mml:msub><mml:mi>ω</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>̲</mml:mo></mml:munder><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="lu-ieq16-3036013.gif"/> </inline-formula> until <inline-formula><tex-math notation="LaTeX">S</tex-math> <mml:math><mml:mi>S</mml:mi></mml:math><inline-graphic xlink:href="lu-ieq17-3036013.gif"/> </inline-formula> becomes <inline-formula><tex-math notation="LaTeX">\lbrace \rbrace</tex-math> <mml:math><mml:mrow><mml:mo>{</mml:mo><mml:mo>}</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="lu-ieq18-3036013.gif"/> </inline-formula>, when more iterations benefit marginally. Due to the potential inconsistency of maximum clique algorithms, we study the problem of maximum clique enumeration and propose an efficient algorithm RMCE to enumerate all maximum cliques in a given graph. As confirmed by the experiments, both RMC and RMCE are much more efficient and robust than previous solutions, RMC can always find the exact maximum clique, and RMCE can always enumerate all maximum cliques in a given graph.
Author Wei, Hao
Yu, Jeffrey Xu
Zhang, Yikai
Lu, Can
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Snippet Cliques refer to subgraphs in an undirected graph such that vertices in each subgraph are pairwise adjacent. The maximum clique problem, to find the clique...
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SubjectTerms Algorithms
Anomalies
Apexes
Approximation algorithms
Bioinformatics
Color
community search
Complex networks
Enumeration
Gene expression
Graph theory
Iterative algorithms
Iterative methods
Lower bounds
Maximum clique
Search problems
Searching
Social networking (online)
Social networks
Upper bound
Upper bounds
Title Enumerating Maximum Cliques in Massive Graphs
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