# Densest k-Subgraph Approximation on Intersection Graphs

@inproceedings{Chen2010DensestKA,
title={Densest k-Subgraph Approximation on Intersection Graphs},
author={Danny Ziyi Chen and Rudolf Fleischer and J. Li},
booktitle={WAOA},
year={2010}
}
• Published in WAOA 9 September 2010
• Mathematics
We study approximation solutions for the densest k-subgraph problem (DS-k) on several classes of intersection graphs. We adopt the concept of σ-quasi elimination orders, introduced by Akcoglu et al. [1], generalizing the perfect elimination orders for chordal graphs, and develop a simple O(σ)-approximation technique for graphs admitting such a vertex order. This concept allows us to derive constant factor approximation algorithms for DS-k on many intersection graph classes, such as chordal…
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## References

SHOWING 1-10 OF 46 REFERENCES
The densest k-subgraph problem on clique graphs
• Mathematics
J. Comb. Optim.
• 2007
This paper considers graphs having special clique graphs, and presents a PTAS for stars of cliques and a dynamic programming algorithm for trees of clique.
On the densest k-subgraph problems
• Mathematics
• 1997
Given an n-vertex graph G and a parameter k, we are to find a k-vertex subgraph with the maximum number of edges. This problem is NP-hard. We show that the problem remains NP-hard even when the
A constant approximation algorithm for the densest k
• Mathematics, Computer Science
Inf. Process. Lett.
• 2008
Clustering and domination in perfect graphs
• Mathematics
Discret. Appl. Math.
• 1984
On the Densest K-subgraph Problem
• Mathematics
• 1997
Given an n-vertex graph G and a parameter k, we are to nd a k-vertex subgraph with the maximum number of edges. This problem is N P-hard. We show that the problem remains N P-hard even when the
Polynomial-time approximation schemes for geometric graphs
• Mathematics, Computer Science
SODA '01
• 2001
These are the first known polynomial-time approximation schemes for NP-hard optimization problems on disk graphs based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible.
Sequential Elimination Graphs ⋆
• Mathematics
• 2009
A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering
On choosing a dense subgraph
• Mathematics, Computer Science
Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science
• 1993
This paper presents a method for converting an approximation algorithm for an unweighted graph problem (from a specific class of maximization problems) into one for the corresponding weighted problem, and apply it to the densest subgraph problem.
Finding Dense Subgraphs with Semidefinite Programming
• Computer Science, Mathematics
APPROX
• 1998
A new approximation algorithm for arbitrary graphs and k=n/c for c > 1 based on semidefinite programming and randomized rounding which achieves for some c the presently best (randomized) approximation factors.