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PTAS for Densest k-Subgraph in Interval Graphs [chapter]

Tim Nonner
2011 Lecture Notes in Computer Science  
Given an interval graph and integer k, we consider the problem of finding a subgraph of size k with a maximum number of induced edges, called densest k-subgraph problem in interval graphs.  ...  We shed light on the approximation complexity of finding a densest k-subgraph in interval graphs by presenting a polynomialtime approximation scheme (PTAS), that is, we show that there is an (1 + ǫ)approximation  ...  There is a PTAS for finding a densest k-subgraph in interval graphs. Proof.  ... 
doi:10.1007/978-3-642-22300-6_53 fatcat:abj6rgh6kfeopag3ora46olrde

PTAS for Densest $$k$$ k -Subgraph in Interval Graphs

Tim Nonner
2014 Algorithmica  
Given an interval graph and integer k, we consider the problem of finding a subgraph of size k with a maximum number of induced edges, called densest k-subgraph problem in interval graphs.  ...  We shed light on the approximation complexity of finding a densest k-subgraph in interval graphs by presenting a polynomialtime approximation scheme (PTAS), that is, we show that there is an (1 + ǫ)approximation  ...  There is a PTAS for finding a densest k-subgraph in interval graphs. Proof.  ... 
doi:10.1007/s00453-014-9956-7 fatcat:5stleolfoze2xoaeuf2bru2tta

Densest k-Subgraph Approximation on Intersection Graphs [chapter]

Danny Z. Chen, Rudolf Fleischer, Jian Li
2011 Lecture Notes in Computer Science  
We study approximation solutions for the densest k-subgraph problem (DS-k) on several classes of intersection graphs.  ...  We also present a polynomial time approximation scheme for the DS-k problem on unit disk graphs using the shifting technique.  ...  The research of this author was supported in part by the US National Science Foundation under Grants CCF-0515203 and CCF-0916606.  ... 
doi:10.1007/978-3-642-18318-8_8 fatcat:bkgw4vghebc7lhyizcmog7xlfi

The densest k-subgraph problem on clique graphs

Maria Liazi, Ioannis Milis, Fanny Pascual, Vassilis Zissimopoulos
2007 Journal of combinatorial optimization  
The Densest k-Subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges.  ...  Especially, towards the elucidation of the open questions concerning the complexity of the problem for interval graphs as well as its approximability for chordal graphs, we consider graphs having special  ...  Acknowledgement The authors would like to thank the two anonymous referees for their valuable comments that significantly improved the presentation.  ... 
doi:10.1007/s10878-007-9069-1 fatcat:kyhvgsl27vgdvabdacu4xqovdq

Page 8504 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
Summary: “Define the density d(G) of a graph G as Seat A polynomial algorithm for finding the densest subgraph of a graph is provided.  ...  ) Determination of the densest subgraph.  ... 

A constant approximation algorithm for the densest k-subgraph problem on chordal graphs

Maria Liazi, Ioannis Milis, Vassilis Zissimopoulos
2008 Information Processing Letters  
The Densest k-Subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges.  ...  In this paper we present a 3-approximation algorithm for the class of chordal graphs. The analysis of our algorithm is based on a graph theoretic lemma of independent interest.  ...  The crucial suggestions of an anonymous referee for the simplification of the proofs of Lemmas 2 and 3 are gratefully acknowledged.  ... 
doi:10.1016/j.ipl.2008.03.016 fatcat:2f4e5ii5hba23kcakgbbhoxu3q

Finding Connected Dense $$k$$ -Subgraphs [chapter]

Xujin Chen, Xiaodong Hu, Changjun Wang
2015 Lecture Notes in Computer Science  
These particularly provide the first non-trivial approximations for the densest connected k-subgraph problem on general graphs.  ...  Given a connected graph G on n vertices and a positive integer k ≤ n, a subgraph of G on k vertices is called a k-subgraph in G.  ...  Several PTAS have been designed for DkSP on unit disk graphs [8] , interval graphs [25] , and a subclass of chordal graphs [24] .  ... 
doi:10.1007/978-3-319-17142-5_22 fatcat:dci2zbr7wzb4ho5vci67viqm24

Finding Dense Subgraphs with Size Bounds [chapter]

Reid Andersen, Kumar Chellapilla
2009 Lecture Notes in Computer Science  
In contrast, we show that damks is nearly as hard to approximate as the densest k-subgraph problem, for which no good approximation algorithm is known.  ...  These problems are relaxed versions of the well-known densest k-subgraph problem (dks), which is to find the densest subgraph with exactly k vertices.  ...  Khot [17] proved there can be no PTAS (polynomial time approximation scheme) for the densest k-subgraph problem, under a reasonable complexity assumption.  ... 
doi:10.1007/978-3-540-95995-3_3 fatcat:hvgzu6ktrbgi3nmtkeqyu3yhse

Finding Connected Dense k-Subgraphs [article]

Xujin Chen, Xiaodong Hu, Changjun Wang
2015 arXiv   pre-print
These particularly provide the first non-trivial approximations for the densest connected k-subgraph problem on general graphs.  ...  We design combinatorial approximation algorithms for finding a connected k-subgraph in G such that its density is at least a factor Ω({n^-2/5,k^2/n^2}) of the density of the densest k-subgraph in G (which  ...  Several PTAS have been designed for DkSP on unit disk graphs [8] , interval graphs [25] , and a subclass of chordal graphs [24] .  ... 
arXiv:1501.07348v1 fatcat:peidahkn4feqjg33wbobfp6jhq

Approximation of the Quadratic Knapsack Problem

Ulrich Pferschy, Joachim Schauer
2016 INFORMS journal on computing  
These hardness of approximation results under certain complexity assumptions carry over from the densest k-subgraph problem.  ...  In this case the quadratic terms of the objective function are not given for each pair of knapsack items.  ...  In any case, our results of Section 3 also imply a PTAS for the densest k-subgraph problem on planar graphs.  ... 
doi:10.1287/ijoc.2015.0678 fatcat:ioqd3xrnofdgtfcvlcvvjoitke

On solving the densest k-subgraph problem on large graphs [article]

Renata Sotirov
2019 arXiv   pre-print
The densest k-subgraph problem is the problem of finding a k-vertex subgraph of a graph with the maximum number of edges.  ...  In order to solve large instances of the densest k-subgraph problem, we introduce two algorithms that are based on the random coordinate descent approach.  ...  The author would like to thank Pavel Dvurechensky for useful discussions on the random coordinate descent approaches.  ... 
arXiv:1901.06344v1 fatcat:djam4i27hne2fcl22ftf7ncacm

Parameterized Complexity of the Sparsest k-Subgraph Problem in Chordal Graphs [chapter]

Marin Bougeret, Nicolas Bousquet, Rodolphe Giroudeau, Rémi Watrigant
2014 Lecture Notes in Computer Science  
We lastly provide an F P T algorithm in interval graphs for Sparsest k-Subgraph, but parameterized by the number of edges of the solution (a stronger parameterization than by k).  ...  In this paper we investigate the parameterized complexity of both Sparsest k-Subgraph and Densest k-Subgraph in chordal graphs.  ...  Still concerning Densest k-Subgraph but in restricted graph classes, [16] developed a PTAS in interval graphs, and [7, 15] developed constant approximation algorithms in chordal graphs.  ... 
doi:10.1007/978-3-319-04298-5_14 fatcat:l7z47o7txjfdte4nq3tvjeejvu

Exact and superpolynomial approximation algorithms for the densest k-subgraph problem

Nicolas Bourgeois, Aristotelis Giannakos, Giorgio Lucarelli, Ioannis Milis, Vangelis Th. Paschos
2017 European Journal of Operational Research  
Moreover, Cai in [15] proved that densest k-subgraph is W[1]-hard, with respect to k even for regular graphs.  ...  graph for the subset which induces the maximum number of edges. densest k-subgraph is a well known optimization problem with various applications as, for example, in the design of public encryption schemes  ...  We would like to thank the anonymous reviewer for her/his valuable comments.  ... 
doi:10.1016/j.ejor.2017.04.034 fatcat:gkovmcae7rfnfftmnoepwipjby

Detecting High Log-Densities -- an O(n^1/4) Approximation for Densest k-Subgraph [article]

Aditya Bhaskara, Moses Charikar, Eden Chlamtac, Uriel Feige and Aravindan Vijayaraghavan
2010 arXiv   pre-print
In the Densest k-Subgraph problem, given a graph G and a parameter k, one needs to find a subgraph of G induced on k vertices that contains the largest number of edges.  ...  We present an algorithm that for every epsilon > 0 approximates the Densest k-Subgraph problem within a ratio of n^(1/4+epsilon) in time n^O(1/epsilon).  ...  Thus the densest 2k-subgraph in the new bipartite graph has at least the same average degree as the densest k-subgraph in G.  ... 
arXiv:1001.2891v1 fatcat:vdxsoa5mmza3xoy6zhmorxmxua

Detecting high log-densities

Aditya Bhaskara, Moses Charikar, Eden Chlamtac, Uriel Feige, Aravindan Vijayaraghavan
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
In the Densest k-Subgraph problem, given a graph G and a parameter k, one needs to find a subgraph of G induced on k vertices that contains the largest number of edges.  ...  We present an algorithm that for every ε > 0 approximates the Densest k-Subgraph problem within a ratio of n 1/4+ε in time n O(1/ε) .  ...  Thus the densest 2k-subgraph in the new bipartite graph has at least the same average degree as the densest k-subgraph in G.  ... 
doi:10.1145/1806689.1806719 dblp:conf/stoc/BhaskaraCCFV10 fatcat:v7edwxr5ujgc7kzitfyaedi6ea
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