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

2011
*
Lecture Notes in Computer Science
*

Given an

doi:10.1007/978-3-642-22300-6_53
fatcat:abj6rgh6kfeopag3ora46olrde
*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. ...##
###
PTAS for Densest $$k$$ k -Subgraph in Interval Graphs

2014
*
Algorithmica
*

Given an

doi:10.1007/s00453-014-9956-7
fatcat:5stleolfoze2xoaeuf2bru2tta
*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. ...##
###
Densest k-Subgraph Approximation on Intersection Graphs
[chapter]

2011
*
Lecture Notes in Computer Science
*

We study approximation solutions

doi:10.1007/978-3-642-18318-8_8
fatcat:bkgw4vghebc7lhyizcmog7xlfi
*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. ...##
###
The densest k-subgraph problem on clique graphs

2007
*
Journal of combinatorial optimization
*

The

doi:10.1007/s10878-007-9069-1
fatcat:kyhvgsl27vgdvabdacu4xqovdq
*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. ...##
###
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

2008
*
Information Processing Letters
*

The

doi:10.1016/j.ipl.2008.03.016
fatcat:2f4e5ii5hba23kcakgbbhoxu3q
*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. ...##
###
Finding Connected Dense $$k$$ -Subgraphs
[chapter]

2015
*
Lecture Notes in Computer Science
*

These particularly provide the first non-trivial approximations

doi:10.1007/978-3-319-17142-5_22
fatcat:dci2zbr7wzb4ho5vci67viqm24
*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] . ...##
###
Finding Dense Subgraphs with Size Bounds
[chapter]

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. ...

##
###
Finding Connected Dense k-Subgraphs
[article]

2015
*
arXiv
*
pre-print

These particularly provide the first non-trivial approximations

arXiv:1501.07348v1
fatcat:peidahkn4feqjg33wbobfp6jhq
*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] . ...##
###
Approximation of the Quadratic Knapsack Problem

2016
*
INFORMS journal on computing
*

These hardness of approximation results under certain complexity assumptions carry over from the

doi:10.1287/ijoc.2015.0678
fatcat:ioqd3xrnofdgtfcvlcvvjoitke
*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*. ...##
###
On solving the densest k-subgraph problem on large graphs
[article]

2019
*
arXiv
*
pre-print

The

arXiv:1901.06344v1
fatcat:djam4i27hne2fcl22ftf7ncacm
*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. ...##
###
Parameterized Complexity of the Sparsest k-Subgraph Problem in Chordal Graphs
[chapter]

2014
*
Lecture Notes in Computer Science
*

We lastly provide an F P T algorithm

doi:10.1007/978-3-319-04298-5_14
fatcat:l7z47o7txjfdte4nq3tvjeejvu
*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*. ...##
###
Exact and superpolynomial approximation algorithms for the densest k-subgraph problem

2017
*
European Journal of Operational Research
*

Moreover, Cai

doi:10.1016/j.ejor.2017.04.034
fatcat:gkovmcae7rfnfftmnoepwipjby
*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. ...##
###
Detecting High Log-Densities -- an O(n^1/4) Approximation for Densest k-Subgraph
[article]

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. ...

##
###
Detecting high log-densities

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. ...

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