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Approximating the Maximum Independent Set and Minimum Vertex Coloring on Box Graphs
[chapter]

*
Algorithmic Aspects in Information and Management
*

We show that for

doi:10.1007/978-3-540-72870-2_32
dblp:conf/aaim/HanIKL07
fatcat:rzl6d5o7ardrfp2hlk4uhvylx4
*box**graphs**on*n vertices which have an*independent**set*of size Ω(n/ log O(1) n)*the**maximum**independent**set*problem can be*approximated*within O(log n/ log log n) in polynomial time. ... Furthermore, we show that*the*chromatic number of a*box**graph**on*n vertices is within an O(log n) factor from*the*size of its*maximum*clique*and*provide an O(log n)*approximation*algorithm for*minimum*... overlap,*one*can generalize*the*divide*and*conquer k-line technique for*maximum**independent**set*(see Section 2) to include*minimum**vertex**coloring*in order to obtain an O(log k+1 n)*approximation*factor ...##
###
Subexponential-Time Algorithms for Maximum Independent Set and Related Problems on Box Graphs
[chapter]

2003
*
Lecture Notes in Computer Science
*

We consider such basic combinatorial problems

doi:10.1007/3-540-45071-8_7
fatcat:hkrtpfpjdjdklhcpkqflivlyq4
*on**box**graphs*as*maximum**independent**set*,*minimum**vertex*cover*and**maximum*induced subgraph with polynomial-time testable hereditary property Π. ... We present an algorithm that solves*maximum**independent**set**and**the*other aforementioned problems in time 2 O(d2 d bn 1−1/d log n)*on*such*box**graphs*in d-dimensions. ... In this paper we study*maximum**independent**set*,*minimum**vertex*cover*and**maximum*induced subgraph with polynomial-time testable hereditary property Π problems*on**the*so called*box**graphs*which are*the*...##
###
The Complexity of Combinatorial Optimization Problems on d‐Dimensional Boxes

2007
*
SIAM Journal on Discrete Mathematics
*

In this paper we prove APX-hardness of

doi:10.1137/050629276
fatcat:lhftgodyfzhfrizfxz75ycv7te
*the**Maximum**Independent**Set*problem in d-*box**graphs*for any fixed d ≥ 3. ...*The**Maximum**Independent**Set*problem in d-*box**graphs*, i.e., in intersection*graphs*of axis-parallel rectangles in R d , is known to be NP-hard for any fixed d ≥ 2. ... This is demonstrated*on**the*problems*Maximum**Independent**Set**and**Minimum**Vertex*Cover in d-*box**graphs*(for any fixed d ≥ 3) proving NP-hardness to achieve an*approximation*factor of 1 + 1 244*and*1 + 1 ...##
###
Hierarchically specified unit disk graphs
[chapter]

1994
*
Lecture Notes in Computer Science
*

These problems include

doi:10.1007/3-540-57899-4_38
fatcat:2g4guthovzejtlyh57exl573vu
*minimum**vertex**coloring*,*maximum**independent**set*,*minimum*clique cover,*minimum*dominating*set**and**minimum**independent*dominating*set*. ... Both PSPACE-hardness results*and*polynomial time*approximations*are presented for most of*the*problems considered. ... Acknowledgements We are indebted to*the*referees for numerous constructive suggestions*on*an earlier draft of this paper. We would like to thank Professors Richard E. Steams*and*Daniel J. ...##
###
Hierarchically specified unit disk graphs

1997
*
Theoretical Computer Science
*

These problems include

doi:10.1016/s0304-3975(96)00008-4
fatcat:ylad3vlzkbbq5aiiwwl5c7jiie
*minimum**vertex**coloring*,*maximum**independent**set*,*minimum*clique cover,*minimum*dominating*set**and**minimum**independent*dominating*set*. ... Both PSPACE-hardness results*and*polynomial time*approximations*are presented for most of*the*problems considered. ... Acknowledgements We are indebted to*the*referees for numerous constructive suggestions*on*an earlier draft of this paper. We would like to thank Professors Richard E. Steams*and*Daniel J. ...##
###
Local approximation schemes for ad hoc and sensor networks

2005
*
Proceedings of the 2005 joint workshop on Foundations of mobile computing - DIALM-POMC '05
*

We present two local approaches that yield polynomial-time

doi:10.1145/1080810.1080827
dblp:conf/dialm/KuhnNMW05
fatcat:kudgjhl63bfc3djufn3h5u6xse
*approximation*schemes (PTAS) for*the**Maximum**Independent**Set**and**Minimum*Dominating*Set*problem in unit disk*graphs*. ...*The*time complexity of both algorithms is O(T MIS + log * n/ε O(1) ), where T MIS is*the*time required to compute a maximal*independent**set*in*the**graph*,*and*n denotes*the*number of nodes. ... For instance, a maximal*independent**set*(MIS) is a constant*approximation*for*minimum*dominating*set**and*for*maximum**independent**set**and*a (∆+1)-*coloring*is a constant*approximation*for*minimum**vertex*...##
###
New approximation algorithms for graph coloring

1994
*
Journal of the ACM
*

*The*previous best upper bound

*on*

*the*number of

*colors*needed for

*coloring*3-

*colorable*n-

*vertex*

*graphs*in polynomial time was 0(~/ -)

*colors*by Berger

*and*Rompel, improvmg a bound of O(w)

*colors*by Wigderson ... This paper explores

*the*

*approximation*problem of

*coloring*k-

*colorable*

*graphs*with as few additional

*colors*as possible m polynomial time, with special focus

*on*

*the*case of k == 3. ...

*Graph*

*coloring*is also closely related to other combinatorial problems such as finding

*the*

*maximum*

*independent*

*set*in a

*graph*(

*the*largest

*set*of vertices such that no two have an edge between them). ...

##
###
Guest Editors' Foreword

2020
*
Discrete & Computational Geometry
*

Geometric Hitting

doi:10.1007/s00454-020-00212-0
fatcat:c7rkm2wq4vax7kdvpn7urjbl7e
*Sets**and**Set*Covers Capacitated covering problems in geometric range spaces are variants of*the**set*cover problem, with constraints*on**the**maximum*number of points assigned to each range ... Sayan Bandyapadhyay, Santanu Bhowmick, Tanmay Inamdar,*and*Kasturi Varadarajan consider covering a*set*of points in R d by*the**minimum*number of balls, subject to capacity constraints for each ball. ...*The**minimum*area of an axis-aligned bounding*box*of such an embedding measures*the*complexity of*the**graph*. ...##
###
Approximability of (Simultaneous) Class Cover for Boxes
[article]

2021
*
arXiv
*
pre-print

*the*smallest cardinality

*set*of axis-aligned

*boxes*which together cover S such that all

*boxes*cover only points of

*the*same

*color*

*and*no

*box*covering a red point intersects a

*box*covering a blue point. ... Bereg et al. (2012) introduced

*the*

*Boxes*Class Cover problem, which has its roots in classification

*and*clustering applications: Given a

*set*of n points in

*the*plane, each

*colored*red or blue, find

*the*... Using

*the*fact that

*Vertex*Cover is NP-hard to

*approximate*within a constant factor of 1 + 1/52

*on*

*graphs*with

*maximum*degree at most 4 [5] we also get

*the*following: Note that as mentionned in

*the*introduction ...

##
###
A Constant Factor Approximation Algorithm for Boxicity of Circular Arc Graphs
[article]

2011
*
arXiv
*
pre-print

Equivalently, it is

arXiv:1102.1544v1
fatcat:a6n3o3gq5zgm5eiqmbtp2wuyz4
*the**minimum*number of interval*graphs**on**the**vertex**set*V such that*the*intersection of their edge*sets*is E. ...*The*time complexity of*the*algorithms to*approximately*compute*the*boxicity is O(mn+n^2) in both these cases*and*in O(mn+kn^2)= O(n^3) time we also get their corresponding*box*representations, where n ... Introduction Boxicity: Boxicity of a*graph*G(V, E) is defined as*the**minimum*number of interval*graphs**on**the**vertex**set*V such that*the*intersection of their edge*sets*is E. ...##
###
Page 2 of Mathematical Reviews Vol. , Issue 90C
[page]

1990
*
Mathematical Reviews
*

*and*, for any two

*colors*c;, ¢2, there are two edges with a common

*vertex*,

*one*

*colored*c;

*and*

*the*other

*colored*c2. ... In particular, optimal partial g-

*colorings*are shown to be equivalent to

*maximum*stable

*sets*

*on*an associated Cartesian sum

*graph*. ...

##
###
Linear-Time Approximation Algorithms for Unit Disk Graphs
[chapter]

2015
*
Lecture Notes in Computer Science
*

Among such results,

doi:10.1007/978-3-319-18263-6_12
fatcat:mx3pi5ibnnht5abnnvbtbntf4u
*the*proposed method yields linear-time (4+eps)-*approximation*for*the**maximum*-weight*independent**set**and**the**minimum*dominating*set*of unit disk*graphs*, thus bringing significant performance ... Numerous*approximation*algorithms for problems*on*unit disk*graphs*have been proposed in*the*literature, exhibiting a sharp trade-off between running times*and**approximation*ratios. ... An extended abstract of this paper appeared in*the*12th Workshop*on**Approximation**and*Online Algorithms (WAOA 2014). ...##
###
Shifting Coresets: Obtaining Linear-Time Approximations for Unit Disk Graphs and Other Geometric Intersection Graphs

2017
*
International journal of computational geometry and applications
*

Among such results,

doi:10.1142/s0218195917500078
fatcat:jckw4etsjrbpnd37ixjbf4iime
*the*proposed method yields linear-time (4 + ε)-*approximations*for*the**maximum*-weight*independent**set**and**the**minimum*dominating*set*of unit disk*graphs*, thus bringing significant performance ... Numerous*approximation*algorithms for problems*on*unit disk*graphs*have been proposed in*the*literature, exhibiting a sharp trade-off between running times*and**approximation*ratios. ... An extended abstract of this paper appeared in*the*12th Workshop*on**Approximation**and*Online Algorithms (WAOA 2014). ...##
###
Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs
[article]

2014
*
arXiv
*
pre-print

paths (EDP) problem

arXiv:1408.0828v1
fatcat:mvahal2h3jbynmxph73fkkn44m
*on*directed acyclic*graphs*(DAGs), where n denotes*the*number of vertices. (3) A tight hardness of packing*vertex*-disjoint k-cycles for large k. (4) An alternative (*and*perhaps simpler ... We resolve some open problems as applications. (1) A tight n^1-ϵ-*approximation*hardness for*the**minimum*consistent deterministic finite automaton (DFA) problem, where n is*the*sample size. ... Acknowledgement: We thank Julia Chuzhoy for suggesting*the*EDP reduction*and*for related discussions when*the*first author was still at*the*University of Chicago. ...##
###
Pre-reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

2014
*
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
*

*The*study of

*graph*... Acknowledgement: We thank Julia Chuzhoy for suggesting

*the*EDP reduction

*and*for related discussions when

*the*first author was still at

*the*University of Chicago. ... . . .]; here, I is a

*graph*which is an instance of a hard

*graph*problem such as

*maximum*

*independent*

*set*or

*minimum*

*coloring*. ...

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