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Doubly Nonnegative and Semidefinite Relaxations for the Densest k-Subgraph Problem

School of Economics and Management, Zhejiang Sci-Tech University, Hangzhou 310018, China
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Entropy 2019, 21(2), 108; https://doi.org/10.3390/e21020108
Received: 19 October 2018 / Revised: 5 January 2019 / Accepted: 19 January 2019 / Published: 24 January 2019
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. View Full-Text
Keywords: densest k-subgraph; doubly nonnegative relaxation; semidefinite relaxation; approximation ratio; NP-hard densest k-subgraph; doubly nonnegative relaxation; semidefinite relaxation; approximation ratio; NP-hard
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Guo, C.-H.; Guo, Y.; Liu, B.-B. Doubly Nonnegative and Semidefinite Relaxations for the Densest k-Subgraph Problem. Entropy 2019, 21, 108.

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