Abstract
We consider the dense subgraph problem that extracts a subgraph with a prescribed number of vertices that has the maximum number of edges (total edge weight in the weighted case) in a given graph. We give approximation algorithms with improved theoretical approximation ratios—assuming that the density of the optimal output subgraph is high, where density is the ratio of number of edges (or sum of edge weights) to the number of edges in the clique on the same number of vertices. Moreover, we investigate the case where the input graph is bipartite, and design a pseudo-polynomial time approximation scheme that can become a PTAS even if the size of the optimal output graph is comparatively small. This is a significant improvement in a theoretical sense, since no constant-ratio approximation algorithm was known previously if the output graph has o(n) vertices.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arora, S., Karger, D., Karpinski, M.: Polynomial Time Approximation Schemes for Dense Instances of NP-hard Problems. In: Proc. STOC 1995, pp. 284–293 (1995)
Asahiro, Y., Hassin, R., Iwama, K.: Complexity of Finding Dense Subgraphs. Discrete Applied Math 121, 15–26 (2002)
Asahiro, Y., Iwama, K., Tamaki, H., Tokuyama, T.: Greedily Finding Dense Subgraphs. Journal of Algorithms 34, 203–221 (2000)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation, Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (1999)
Czygrinow, A.: Maximum Dispersion Problem in Dense Graphs. Operation Research Letters 27, 223–227 (2000)
Feige, U.: Relations between Average Case Complexity and Approximation Complexity. In: Proc. STOC 2002, pp. 534–543 (2002)
Feige, U., Kortsarz, G., Peleg, D.: The Dense k-Subgraph Problem. Algorithmica 29, 410–421 (2001)
Hochbaum, D.S.: Approximating clique and biclique problems. J. Algorithms 29, 174–200 (1998)
Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A Fast Parametric Maximum Flow Algorithm and Applications. SIAM J. Comput. 18, 30–55 (1989)
Kortsarz, G., Peleg, D.: On Choosing a Dense Subgraph. In: Proc. FOCS 1993, pp. 692–701 (1993)
Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Suzuki, A., Tokuyama, T. (2005). Dense Subgraph Problems with Output-Density Conditions. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_28
Download citation
DOI: https://doi.org/10.1007/11602613_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30935-2
Online ISBN: 978-3-540-32426-3
eBook Packages: Computer ScienceComputer Science (R0)