Parameterized algorithms for Edge Biclique and related problems

Qi, Feng; Li, Shaohua; Zhou, Zeyang; Wang, Jianxin

doi:10.1016/j.tcs.2017.09.027

Cited by 9 publications

(3 citation statements)

References 18 publications

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“…Maximum Biclique Search and its Variants. The maximum biclique problem has become increasingly popular in recent years [30,29,7]. [30] proposes an integer programming methodology to find the maximum biclique in general graphs.…”

Section: Related Workmentioning

confidence: 99%

“…[29] develops a Monte Carlo algorithm for extracting a list of maximal bicliques, which contains a maximum biclique with fixed probability. [7] studies the parameterized maximum biclique problem in bipartite graphs, that reports if there exists a biclique with at least k edges, where k is a given integer parameter. Besides, there are two variants of the maximum biclique problem, i.e., the maximum vertex biclique and the maximum balanced biclique.…”

Section: Related Workmentioning

confidence: 99%

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Maximum biclique search at billion scale

Lyu

Lin

et al. 2020

Proc. VLDB Endow.

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Maximum biclique search, which finds the biclique with the maximum number of edges in a bipartite graph, is a fundamental problem with a wide spectrum of applications in different domains, such as E-Commerce, social analysis, web services, and bioinformatics. Unfortunately, due to the difficulty of the problem in graph theory, no practical solution has been proposed to solve the issue in large-scale real-world datasets. Existing techniques for maximum clique search on a general graph cannot be applied because the search objective of maximum biclique search is two-dimensional, i.e., we have to consider the size of both parts of the biclique simultaneously. In this paper, we divide the problem into several subproblems each of which is specified using two parameters. These subproblems are derived in a progressive manner, and in each subproblem we can restrict the search in a very small part of the original bipartite graph. We prove that a logarithmic number of subproblems is enough to guarantee the algorithm correctness. To minimize the computational cost, we show how to reduce significantly the bipartite graph size for each subproblem while preserving the maximum biclique satisfying certain constraints by exploring the properties of one-hop and two-hop neighbors for each vertex. We use several real datasets from various application domains, one of which contains over 300 million vertices and 1.3 billion edges, to demonstrate the high efficiency and scalability of our proposed solution. It is reported that 50% improvement on recall can be achieved after applying our method in Alibaba Group to identify the fraudulent transactions in their e-commerce networks. This further demonstrates the usefulness of our techniques in practice.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Maximum biclique search at billion scale

Lyu

Lin

et al. 2020

Proc. VLDB Endow.

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show abstract

“…Non-Induced Max-Edge Biclique can be solved by solving √ k instances of Non-Induced (k 1 , k 2 )-Biclique and thus the latter problem can be considered to be more difficult in our setting. Non-Induced Max-Edge Biclique can be solved in O(k 2.5 k √ k n) time by applying the algorithm for Induced Max-Edge Biclique on bipartite graphs [10].…”

Section: And St ∈ E(g)mentioning

confidence: 99%

Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

Koana¹,

Komusiewicz²,

Sommer³

2020

Preprint

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A graph G is weakly γ-closed if every induced subgraph of G contains one vertex v such that for each non-neighbor u of v it holds that |N (u) ∩ N (v)| < γ. The weak closure γ(G) of a graph, recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number such that G is weakly γ-closed. This graph parameter is never larger than the degeneracy (plus one) and can be significantly smaller. Extending the work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that several problems related to finding dense subgraphs, such as the enumeration of bicliques and s-plexes, are fixed-parameter tractable with respect to γ(G). Moreover, we show that the problem of determining whether a weakly-γ-closed graph G has a subgraph on at least k vertices that belongs to a graph class G which is closed under taking subgraphs admits a kernel with at most γk 2 vertices. Finally, we provide fixed-parameter algorithms for Independent Dominating Set and Dominating Clique when parameterized by γ + k where k is the solution size.

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Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

2023

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A graph G is weakly $$\gamma $$ γ -closed if every induced subgraph of G contains one vertex v such that for each non-neighbor u of v it holds that $$ \vert N(u)\cap N(v) \vert <\gamma $$ | N ( u ) ∩ N ( v ) | < γ . The weak closure $$\gamma (G)$$ γ ( G ) of a graph, recently introduced by Fox et al. (SIAM J Comput 49(2):448–464, 2020), is the smallest number such that G is weakly $$\gamma $$ γ -closed. This graph parameter is never larger than the degeneracy (plus one) and can be significantly smaller. Extending the work of Fox et al. (2020) on clique enumeration, we show that several problems related to finding dense subgraphs, such as the enumeration of bicliques and s-plexes, are fixed-parameter tractable with respect to $$\gamma (G)$$ γ ( G ) . Moreover, we show that the problem of determining whether a weakly $$\gamma $$ γ -closed graph G has a subgraph on at least k vertices that belongs to a graph class $$\mathcal {G}$$ G which is closed under taking subgraphs admits a kernel with at most $$\gamma k^2$$ γ k 2 vertices. Finally, we provide fixed-parameter algorithms for Independent Dominating Set and Dominating Clique when parameterized by $$\gamma +k$$ γ + k where k is the solution size. Furthermore, we show that Independent Dominating Set does not admit a polynomial kernel for constant $$\gamma $$ γ under standard assumptions.

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Parameterized algorithms for Edge Biclique and related problems

Cited by 9 publications

References 18 publications

Maximum biclique search at billion scale

Maximum biclique search at billion scale

Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

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