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Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs

2017
*
Algorithmica
*

1 2 3 4 4 + 3 + 2 + 1 = 10

doi:10.1007/s00453-017-0362-9
fatcat:mwfvkxphtbgijkr2yzle3vsfee
*Max*-*Coloring**Interval**Graphs*: Given*a*weighted*interval**graph*G , find*a**coloring*σ of G that minimizes C ∈σ w C where w C :=*max*I ∈C w I Non-Uniform Resource Requirements ... 1 2 3 4 4 + 4 + 3 = 11*Max*-*Coloring**Interval**Graphs*: Given*a*weighted*interval**graph*G , find*a**coloring*σ of G that minimizes C ∈σ w C where w C :=*max*I ∈C w I Non-Uniform Resource Requirements What ... Assume that m = 2 and κ = 2 cost increase = w m m ǫ 2 ω log n = ǫ 2 log n · wm w 1 m · w 1 ω < ǫ · O(1) · OPT*clusters*in P and still the cost of σ is ≤ (1 + ǫ)OPT. ...##
###
Clique Clustering Yields a PTAS for max-Coloring Interval Graphs
[chapter]

2011
*
Lecture Notes in Computer Science
*

This problem, called

doi:10.1007/978-3-642-22006-7_16
fatcat:ekqsovxiy5eknhfgdoylyxggee
*max*-*coloring**interval**graphs*, contains the classical problem of*coloring**interval**graphs*as*a*special case*for*uniform weights, and it arises in many practical scenarios such as memory ... , which we call*clique**clustering*. ... There is*a**PTAS**for**max*-*coloring**interval**graphs*. Proof. Let P be*a**clique*partition as proposed by Lemma 2. ...##
###
Minimum Clique Partition in Unit Disk Graphs

2011
*
Graphs and Combinatorics
*

We present two improved approximation algorithms

doi:10.1007/s00373-011-1026-1
fatcat:zvh7wt5ljzb6nhohuqxgdqydxu
*for*minimum*clique*partition in unit disk*graphs*: (I)*A*polynomial time approximation scheme (*PTAS*) running in time n O(1/ε 2 ) . ... The minimum*clique*partition (MCP) problem is that of partitioning the vertex set of*a*given*graph*into*a*minimum number of*cliques*. ... Mutual proximity of nodes in*a**cluster*is therefore*a*key criterion, and*cliques*as*clusters*describe it the best. ...##
###
Minimum clique partition in unit disk graphs
[article]

2009
*
arXiv
*
pre-print

We present two improved approximation algorithms

arXiv:0909.1552v1
fatcat:oplcp4ctn5bcrioa25bg3oluum
*for*minimum*clique*partition in unit disk*graphs*: (I)*A*polynomial time approximation scheme (*PTAS*) running in time n^O(1/^2). ... The minimum*clique*partition (MCP) problem is that of partitioning the vertex set of*a*given*graph*into*a*minimum number of*cliques*. ... Mutual proximity of nodes in*a**cluster*is therefore*a*key criterion, and*cliques*as*clusters*describe it the best. ...##
###
Approximate MAP Estimation for Pairwise Potentials via Baker's Technique
[article]

2018
*
arXiv
*
pre-print

We show that computing

arXiv:1412.0340v4
fatcat:bwu3jpcq3fc5bgpugzpvmfwnri
*MAX*2-CSP admits polynomial-time approximation scheme (*PTAS*) on planar*graphs*,*graphs*with bounded local treewidth, H-minor-free*graphs*, geometric*graphs*with bounded density and ... Our method also gives the first*PTAS**for*computing the ground state of ferromagnetic Edwards-Anderson model without external magnetic field on d-dimensional lattice*graphs*. ... If the*graph*G is x-*colorable*, then the value of the minsum S min = min σ i∈V f i must fall into the*interval*[w x , w x · n]. ...##
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On the complexity of the selective graph coloring problem in some special classes of graphs

2014
*
Theoretical Computer Science
*

Given an integer k ≥ 1 and

doi:10.1016/j.tcs.2013.04.018
fatcat:bg3vx2zdo5enpiocal22u3wihi
*a**graph*G = (V, E) with*a*partition V 1 , . . . , V p of V , it consists in deciding whether there exists*a*set V * in G such that V * ∩ V i = 1*for*all i ∈ {1, . . . , p}, and ... such that the*graph*induced by V * is k-*colorable*. ... Marc Demange also acknowledges the support by the French Agency*for*Research under the DEFIS program TODO, ANR-09-EMER-010. ...##
###
A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

Our main result is

doi:10.1007/978-3-642-13731-0_19
fatcat:hsqhdfy6arglhd7gtzdg3ycq6a
*a*weakly robust polynomial time approximation scheme (*PTAS*)*for*UDGs expressed with edge-lengths, it either (i) computes*a**clique*partition or (ii) gives*a*certificate that the*graph*... We consider the problem of partitioning the set of vertices of*a*given unit disk*graph*(UDG) into*a*minimum number of*cliques*. ... Acknowledgments We thank Sriram Pemmaraju, Lorna Stewart, and Zoya Svitkina*for*helpful discussions. Our thanks to an anonymous source*for*pointing out the result of Capoyleas et al. [4] . ...##
###
A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

2011
*
Algorithmica
*

Our main result is

doi:10.1007/s00453-011-9503-8
fatcat:atjx2d4nzzbehej7pyxtryz6fe
*a*weakly robust polynomial time approximation scheme (*PTAS*)*for*UDGs expressed with edge-lengths, it either (i) computes*a**clique*partition or (ii) gives*a*certificate that the*graph*... We consider the problem of partitioning the set of vertices of*a*given unit disk*graph*(UDG) into*a*minimum number of*cliques*. ... Acknowledgments We thank Sriram Pemmaraju, Lorna Stewart, and Zoya Svitkina*for*helpful discussions. Our thanks to an anonymous source*for*pointing out the result of Capoyleas et al. [4] . ...##
###
Diversity maximization in doubling metrics
[article]

2018
*
arXiv
*
pre-print

Finally, we provide

arXiv:1809.09521v1
fatcat:mnw2vn7jqfcdvdcw47t52u2lme
*a*proof of NP-hardness*for*remote-*clique*with squared distances in doubling metric spaces. ... The existence of such*a**PTAS*however was left open. ... [3] that*for*any constants 0 < ε ≤ 2 3 and δ > 0, it is hard to distinguish between*a*) an n-vertex*graph*containing*a**clique*of size k = n 1−ε , and b) an n-vertex*graph*where each k-vertex subset induces ...##
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Independent Sets of Dynamic Rectangles: Algorithms and Experiments
[article]

2020
*
arXiv
*
pre-print

*For*uniform rectangles, we first give an algorithm that maintains

*a*4-approximate

*MAX*-IS with O(1) update time. ... We present the first deterministic algorithm

*for*maintaining

*a*MIS (and thus

*a*4-approximate

*MAX*-IS) of

*a*dynamic set of uniform rectangles with amortized sub-logarithmic update time. ... Moreover, the

*Max*-IS problem admits

*a*polynomial-time approximation scheme (

*PTAS*)

*for*unit square intersection

*graphs*[20] and pseudo disks [14] . ...

##
###
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
[article]

2016
*
arXiv
*
pre-print

We prove that such

arXiv:1501.00721v2
fatcat:5y2riuxz4fbhrc3u7nen5yicxy
*graphs*have small separators. Next, we present efficient (1+ε)-approximation algorithms*for*these*graphs*,*for*Independent Set, Set Cover, and Dominating Set problems, among others. ... We also prove corresponding hardness of approximation*for*some of these optimization problems, providing*a*characterization of their intractability in terms of density. ... We are particularly grateful to the anonymous referee of*a*previous version of this paper who pointed out the connection of our work to*graphs*with bounded expansion. ...##
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Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

2017
*
SIAM journal on computing (Print)
*

*For*example, the new algorithms

*yield*the only

*PTAS*known

*for*covering points by fat triangles (that are shallow). ... This family is quite general, includes planar

*graphs*, and in particular is

*a*subset of the family of

*graphs*that have polynomial expansion. ... The authors thank Mark de Berg

*for*useful discussions related to the problems studied in this paper. In particular, he pointed out an improved bound

*for*Lemma 2.2.7 p9 , detailed in the Appendix

*A*.2. ...

##
###
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
[chapter]

2015
*
Lecture Notes in Computer Science
*

*For*example, the new algorithms

*yield*the only

*PTAS*known

*for*covering points by fat triangles (that are shallow). ... This family is quite general, includes planar

*graphs*, and in particular is

*a*subset of the family of

*graphs*that have polynomial expansion. ... The authors thank Mark de Berg

*for*useful discussions related to the problems studied in this paper. In particular, he pointed out an improved bound

*for*Lemma 2.2.7 p9 , detailed in the Appendix

*A*.2. ...

##
###
Flexible Resource Allocation for Clouds and All-Optical Networks
[article]

2016
*
arXiv
*
pre-print

We further show that the contiguous variant admits

arXiv:1605.04982v1
fatcat:nckytedfz5gyfllcs5zpbzfjsy
*a*(5/4 + ε)-approximation algorithm,*for*any fixed ε > 0, on instances whose job*intervals*form*a*proper*interval**graph*. ... to competing job*intervals*in*a*flexible manner. ... Acknowledgments: We thank Magnús Halldórsson and Viswanath Nagarajan*for*valuable discussions. ...##
###
Real-time scheduling to minimize machine busy times

2014
*
Journal of Scheduling
*

Better bounds and exact algorithms are derived

doi:10.1007/s10951-014-0411-z
fatcat:2gclx6xu2rfj3fwqe235johvhi
*for*several special cases, including proper*interval**graphs*,*intervals*forming*a**clique*and laminar families of*intervals*. ... We are given*a*set of jobs, each having*a*release time,*a*due date,*a*processing time and demand*for*machine capacity. ... Acknowledgment: We thank Dor Arad*for*helpful discussions on an earlier version of the paper. We are also grateful to the two anonymous referees*for*valuable comments and suggestions. ...
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