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Approximating the Maximum Independent Set and Minimum Vertex Coloring on Box Graphs
[chapter]
Algorithmic Aspects in Information and Management
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We show that for 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 ...
doi:10.1007/978-3-540-72870-2_32
dblp:conf/aaim/HanIKL07
fatcat:rzl6d5o7ardrfp2hlk4uhvylx4
Subexponential-Time Algorithms for Maximum Independent Set and Related Problems on Box Graphs
[chapter]
2003
Lecture Notes in Computer Science
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We consider such basic combinatorial problems 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 ...
doi:10.1007/3-540-45071-8_7
fatcat:hkrtpfpjdjdklhcpkqflivlyq4
The Complexity of Combinatorial Optimization Problems on d‐Dimensional Boxes
2007
SIAM Journal on Discrete Mathematics
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In this paper we prove APX-hardness of 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 ...
doi:10.1137/050629276
fatcat:lhftgodyfzhfrizfxz75ycv7te
Hierarchically specified unit disk graphs
[chapter]
1994
Lecture Notes in Computer Science
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These problems include 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. ...
doi:10.1007/3-540-57899-4_38
fatcat:2g4guthovzejtlyh57exl573vu
Hierarchically specified unit disk graphs
1997
Theoretical Computer Science
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These problems include 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. ...
doi:10.1016/s0304-3975(96)00008-4
fatcat:ylad3vlzkbbq5aiiwwl5c7jiie
Local approximation schemes for ad hoc and sensor networks
2005
Proceedings of the 2005 joint workshop on Foundations of mobile computing - DIALM-POMC '05
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We present two local approaches that yield polynomial-time 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 ...
doi:10.1145/1080810.1080827
dblp:conf/dialm/KuhnNMW05
fatcat:kudgjhl63bfc3djufn3h5u6xse
New approximation algorithms for graph coloring
1994
Journal of the ACM
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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). ...
doi:10.1145/176584.176586
fatcat:lzqt4uzmt5darjecadshe57hri
Guest Editors' Foreword
2020
Discrete & Computational Geometry
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Geometric Hitting 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. ...
doi:10.1007/s00454-020-00212-0
fatcat:c7rkm2wq4vax7kdvpn7urjbl7e
Approximability of (Simultaneous) Class Cover for Boxes
[article]
2021
arXiv
pre-print
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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 ...
arXiv:2106.12969v1
fatcat:4sfmubltirbhtftbff7drenlqu
A Constant Factor Approximation Algorithm for Boxicity of Circular Arc Graphs
[article]
2011
arXiv
pre-print
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Equivalently, it is 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. ...
arXiv:1102.1544v1
fatcat:a6n3o3gq5zgm5eiqmbtp2wuyz4
Page 2 of Mathematical Reviews Vol. , Issue 90C
[page]
1990
Mathematical Reviews
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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
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Among such results, 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). ...
doi:10.1007/978-3-319-18263-6_12
fatcat:mx3pi5ibnnht5abnnvbtbntf4u
Shifting Coresets: Obtaining Linear-Time Approximations for Unit Disk Graphs and Other Geometric Intersection Graphs
2017
International journal of computational geometry and applications
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Among such results, 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). ...
doi:10.1142/s0218195917500078
fatcat:jckw4etsjrbpnd37ixjbf4iime
Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs
[article]
2014
arXiv
pre-print
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paths (EDP) problem 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. ...
arXiv:1408.0828v1
fatcat:mvahal2h3jbynmxph73fkkn44m
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
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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. ...
doi:10.1109/focs.2014.54
dblp:conf/focs/ChalermsookLN14
fatcat:wfkcavnby5hwpov2dvpksgsiyq
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