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Keywords = densest subgraph

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9 pages, 236 KB  
Article
Algorithms for Densest Subgraphs of Vertex-Weighted Graphs
by Zhongling Liu, Wenbin Chen, Fufang Li, Ke Qi and Jianxiong Wang
Mathematics 2024, 12(14), 2206; https://doi.org/10.3390/math12142206 - 14 Jul 2024
Viewed by 1481
Abstract
Finding the densest subgraph has tremendous potential in computer vision and social network research, among other domains. In computer vision, it can demonstrate essential structures, and in social network research, it aids in identifying closely associated communities. The densest subgraph problem is finding [...] Read more.
Finding the densest subgraph has tremendous potential in computer vision and social network research, among other domains. In computer vision, it can demonstrate essential structures, and in social network research, it aids in identifying closely associated communities. The densest subgraph problem is finding a subgraph with maximum mean density. However, most densest subgraph-finding algorithms are based on edge-weighted graphs, where edge weights can only represent a single value dimension, whereas practical applications involve multiple dimensions. To resolve the challenge, we propose two algorithms for resolving the densest subgraph problem in a vertex-weighted graph. First, we present an exact algorithm that builds upon Goldberg’s original algorithm. Through theoretical exploration and analysis, we rigorously verify our proposed algorithm’s correctness and confirm that it can efficiently run in polynomial time O(n(n + m)log2n) is its temporal complexity. Our approach can be applied to identify closely related subgroups demonstrating the maximum average density in real-life situations. Additionally, we consistently offer an approximation algorithm that guarantees an accurate approximation ratio of 2. In conclusion, our contributions enrich theoretical foundations for addressing the densest subgraph problem. Full article
(This article belongs to the Special Issue Mathematical and Computing Sciences for Artificial Intelligence)
12 pages, 1788 KB  
Article
Augmentation of Densest Subgraph Finding Unsupervised Feature Selection Using Shared Nearest Neighbor Clustering
by Deepesh Chugh, Himanshu Mittal, Amit Saxena, Ritu Chauhan, Eiad Yafi and Mukesh Prasad
Algorithms 2023, 16(1), 28; https://doi.org/10.3390/a16010028 - 3 Jan 2023
Cited by 3 | Viewed by 2267
Abstract
Determining the optimal feature set is a challenging problem, especially in an unsupervised domain. To mitigate the same, this paper presents a new unsupervised feature selection method, termed as densest feature graph augmentation with disjoint feature clusters. The proposed method works in two [...] Read more.
Determining the optimal feature set is a challenging problem, especially in an unsupervised domain. To mitigate the same, this paper presents a new unsupervised feature selection method, termed as densest feature graph augmentation with disjoint feature clusters. The proposed method works in two phases. The first phase focuses on finding the maximally non-redundant feature subset and disjoint features are added to the feature set in the second phase. To experimentally validate, the efficiency of the proposed method has been compared against five existing unsupervised feature selection methods on five UCI datasets in terms of three performance criteria, namely clustering accuracy, normalized mutual information, and classification accuracy. The experimental analyses have shown that the proposed method outperforms the considered methods. Full article
(This article belongs to the Special Issue Algorithms for Feature Selection)
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22 pages, 918 KB  
Article
Subgraph Query Matching in Multi-Graphs Based on Node Embedding
by Muhammad Anwar, Aboul Ella Hassanien, Václav Snás̃el and Sameh H. Basha
Mathematics 2022, 10(24), 4830; https://doi.org/10.3390/math10244830 - 19 Dec 2022
Cited by 1 | Viewed by 3549
Abstract
This paper presents an efficient algorithm for matching subgraph queries in a multi-graph based on features-based indexing techniques. The KD-tree data structure represents these nodes’ features, while the set-trie index data structure represents the multi-edges to make queries effectively. The vertex core number, [...] Read more.
This paper presents an efficient algorithm for matching subgraph queries in a multi-graph based on features-based indexing techniques. The KD-tree data structure represents these nodes’ features, while the set-trie index data structure represents the multi-edges to make queries effectively. The vertex core number, triangle number, and vertex degree are the eight features’ main features. The densest vertex in the query graph is extracted based on these main features. The proposed model consists of two phases. The first phase’s main idea is that, for the densest extracted vertex in the query graph, find the density similar neighborhood structure in the data graph. Then find the k-nearest neighborhood query to obtain the densest subgraph. The second phase for each layer graph, mapping the vertex to feature vector (Vertex Embedding), improves the proposed model. To reduce the node-embedding size to be efficient with the KD-tree, indexing a dimension reduction, the principal component analysis (PCA) method is used. Furthermore, symmetry-breaking conditions will remove the redundancy in the generated pattern matching with the query graph. In both phases, the filtering process is applied to minimize the number of candidate data nodes of the initiate query vertex. The filtering process is applied to minimize the number of candidate data nodes of the initiate query vertex. Finally, testing the effect of the concatenation of the structural features (orbits features) with the meta-features (summary of general, statistical, information-theoretic, etc.) for signatures of nodes on the model performance. The proposed model is tested over three real benchmarks, multi-graph datasets, and two randomly generated multi-graph datasets. The results agree with the theoretical study in both random cliques and Erdos random graph. The experiments showed that the time efficiency and the scalability results of the proposed model are acceptable. Full article
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18 pages, 270 KB  
Article
In Search of the Densest Subgraph
by András Faragó and Zohre R. Mojaveri
Algorithms 2019, 12(8), 157; https://doi.org/10.3390/a12080157 - 2 Aug 2019
Cited by 10 | Viewed by 4792
Abstract
In this survey paper, we review various concepts of graph density, as well as associated theorems and algorithms. Our goal is motivated by the fact that, in many applications, it is a key algorithmic task to extract a densest subgraph from an input [...] Read more.
In this survey paper, we review various concepts of graph density, as well as associated theorems and algorithms. Our goal is motivated by the fact that, in many applications, it is a key algorithmic task to extract a densest subgraph from an input graph, according to some appropriate definition of graph density. While this problem has been the subject of active research for over half of a century, with many proposed variants and solutions, new results still continuously emerge in the literature. This shows both the importance and the richness of the subject. We also identify some interesting open problems in the field. Full article
15 pages, 369 KB  
Article
Doubly Nonnegative and Semidefinite Relaxations for the Densest k-Subgraph Problem
by Chuan-Hao Guo, Yuan Guo and Bei-Bei Liu
Entropy 2019, 21(2), 108; https://doi.org/10.3390/e21020108 - 24 Jan 2019
Viewed by 2909
Abstract
The densest k-subgraph (DkS) maximization problem is to find a set of k vertices with maximum total weight of edges in the subgraph induced by this set. This problem is in general NP-hard. In this paper, two relaxation methods for solving the [...] Read more.
The densest k-subgraph (DkS) maximization problem is to find a set of k vertices with maximum total weight of edges in the subgraph induced by this set. This problem is in general NP-hard. In this paper, two relaxation methods for solving the DkS problem are presented. One is doubly nonnegative relaxation, and the other is semidefinite relaxation with tighter relaxation compare with the relaxation of standard semidefinite. The two relaxation problems are equivalent under the suitable conditions. Moreover, the corresponding approximation ratios’ results are given for these relaxation problems. Finally, some numerical examples are tested to show the comparison of these relaxation problems, and the numerical results show that the doubly nonnegative relaxation is more promising than the semidefinite relaxation for solving some DkS problems. Full article
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22 pages, 848 KB  
Article
Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis
by Pasin Manurangsi
Algorithms 2018, 11(1), 10; https://doi.org/10.3390/a11010010 - 17 Jan 2018
Cited by 42 | Viewed by 7198
Abstract
The Small Set Expansion Hypothesis is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose (edge) expansion is almost zero and one in which all small subsets of vertices have expansion [...] Read more.
The Small Set Expansion Hypothesis is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose (edge) expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove conditional inapproximability results with essentially optimal ratios for the following graph problems based on this hypothesis: Maximum Edge Biclique, Maximum Balanced Biclique, Minimum k-Cut and Densest At-Least-k-Subgraph. Our hardness results for the two biclique problems are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani to avoid locality of gadget reductions with a generalization of Bansal and Khot’s long code test whereas our results for Minimum k-Cut and Densest At-Least-k-Subgraph are shown via elementary reductions. Full article
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