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}
}
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… 
Approximating the Sparsestk-Subgraph in Chordal Graphs
TLDR
This work investigates Sparsestk-Subgraph in graph classes where independent set is polynomial-time solvable, such as subclasses of perfect graphs, and shows the 𝓝𝓟-hardness of SparsESTk- subgraph on chordal graphs and a greedy 2-approximation algorithm.
PTAS for Densest k-Subgraph in Interval Graphs
  • T. Nonner
  • Mathematics, Computer Science
    WADS
  • 2011
TLDR
It is shown that there is an (1+e)-approximation algorithm for any e > 0, which is the first such approximation scheme for the densest k-subgraph problem in an important graph class without any further restrictions.
Approximating the Sparsest k-Subgraph in Chordal Graphs
TLDR
This work investigates Sparsest k -Subgraph in graph classes where independent set is polynomial-time solvable, such as subclasses of perfect graphs, and shows the \(\mathcal{NP}\)-hardness of SparsEST k-Subgraph on chordal graphs,and a greedy 2-approximation algorithm.
Parameterized Complexity of the Sparsest k-Subgraph Problem in Chordal Graphs
TLDR
This paper provides simple proofs that Densest k-Subgraph in chordal graphs is FPT and does not admit a polynomial kernel unless \({\mathcal NP} \subseteq co- NP/poly\) (both parameterized by k).
The k-Sparsest Subgraph Problem
TLDR
This paper uses dynamic programming to design a PTAS in proper interval graph and an FPT algorithm in interval graphs (parameterized by the number of edges in the solution) and presents a simple greedy tight 2-approximation algorithm in Proper interval graphs.
Parameterized Complexity of the Sparsest k-Subgraph in Chordal Graphs
(This is the long version of the paper accepted to SOFSEM 2014) In this paper we study the Sparsest k-Subgraph problem which consists in finding a subset of k vertices in a graph which induces the
NP-hardness of k-sparsest subgraph in Chordal Graphs
Given a simple undirected graph G = (V,E) and an integer k ≤ |V|, the k-sparsest subgraph problem asks for a set of k vertices which induce the minimum number of edges. Whereas its special case
Approximating the Quadratic Knapsack Problem on Special Graph Classes
TLDR
It is shown that QKP permits an FPTAS on graphs of bounded treewidth and a PTAS on planar graphs and more generally on H-minor free graphs and that the problem might have a bad approximability behaviour on all graph classes containing large cliques.
...
...

References

SHOWING 1-10 OF 46 REFERENCES
The densest k-subgraph problem on clique graphs
TLDR
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
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
Clustering and domination in perfect graphs
On the Densest K-subgraph Problem
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
TLDR
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 ⋆
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
  • G. Kortsarz, D. Peleg
  • Mathematics, Computer Science
    Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science
  • 1993
TLDR
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
TLDR
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.
...
...