Abstract
The density of a subgraph in an undirected graph is the sum of the subgraph’s edge weights divided by the number of the subgraph’s vertices. Finding an induced subgraph of maximum density among all subgraphs with at least k vertices is called as the densest at-least-k-subgraph problem (DalkS).
In this paper, we first present a polynomial time algorithms for DalkS when k is bounded by some constant c. For a graph of n vertices and m edges, our algorithm is of time complexity \(O(n^{c + 3} \log n)\), which improve previous best time complexity \(O(n^c(n+m)^{4.5})\).
Second, we give a greedy approximation algorithm for the Densest Subgraph with a Specified Subset Problem. We show that the greedy algorithm is of approximation ratio \(2\cdot (1+ \frac{k}{3})\), where k is the element number of the specified subset.
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Acknowledgments
We would like to thank the anonymous referees for their careful readings of the manuscripts and many useful suggestions.sparabreak Wenbin Chen’s research has been supported by the National Science Foundation of China (NSFC) under Grant No. 11271097. Lingxi Peng’s research has been partly supported by the Funding Program for Research Development in Institutions of Higher Learning Under the Jurisdiction of Guangzhou Municipality under Grant No. 2012A077. Jianxiong Wang’s research was partially supported under Foundation for Distinguished Young Talents in Higher Education of Guangdong (2012WYM0105 and 2012LYM0105) and Funding Program for Research Development in Institutions of Higher Learning Under the Jurisdiction of Guangzhou Municipality (2012A143). FuFang Li’s work had been co-financed by: Natural Science Foundation of China under Grant No. 61472092; Guangdong Provincial Science and Technology Plan Project under Grant No. 2013B010401037; and GuangZhou Municipal High School Science Research Fund under grant No. 1201421317. Maobin Tang’s research has been supported under Guangdong Province’s Science and Technology Projects under Grant No. 2012A020602065 and the research project of Guangzhou education bureau under Grant No. 2012A075.
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Chen, W., Peng, L., Wang, J., Li, F., Tang, M. (2015). Algorithms for the Densest Subgraph with at Least k Vertices and with a Specified Subset. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_41
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