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

Xin Han, Kazuo Iwama, Rolf Klein, Andrzej Lingas
Algorithmic Aspects in Information and Management
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  ...

### Subexponential-Time Algorithms for Maximum Independent Set and Related Problems on Box Graphs [chapter]

Andrzej Lingas, Martin Wahlen
2003 Lecture Notes in Computer Science
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  ...

### The Complexity of Combinatorial Optimization Problems on d‐Dimensional Boxes

Miroslav Chlebík, Janka Chlebíková
2007 SIAM Journal on Discrete Mathematics
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  ...

### Hierarchically specified unit disk graphs [chapter]

M. V. Marathe, V. Radhakrishnan, H. B. Hunt, S. S. Ravi
1994 Lecture Notes in Computer Science
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.  ...

### Hierarchically specified unit disk graphs

1997 Theoretical Computer Science
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.  ...

### Local approximation schemes for ad hoc and sensor networks

Fabian Kuhn, Tim Nieberg, Thomas Moscibroda, Roger Wattenhofer
2005 Proceedings of the 2005 joint workshop on Foundations of mobile computing - DIALM-POMC '05
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  ...

### New approximation algorithms for graph coloring

Avrim Blum
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

Bettina Speckmann, Csaba D. Tóth
2020 Discrete & Computational Geometry
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.  ...

### Approximability of (Simultaneous) Class Cover for Boxes [article]

Jean Cardinal, Justin Dallant, John Iacono
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]

Abhijin Adiga and Jasine Babu and L. Sunil Chandran
2011 arXiv   pre-print
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.  ...

### 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]

Guilherme D. da Fonseca, Vinícius G. Pereira de Sá, Celina M. H. de Figueiredo
2015 Lecture Notes in Computer Science
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).  ...

### Shifting Coresets: Obtaining Linear-Time Approximations for Unit Disk Graphs and Other Geometric Intersection Graphs

Guilherme D. da Fonseca, Vinícius Gusmão Pereira de Sá, Celina Miraglia Herrera de Figueiredo
2017 International journal of computational geometry and applications
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).  ...

### Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs [article]

Parinya Chalermsook, Bundit Laekhanukit, Danupon Nanongkai
2014 arXiv   pre-print
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.  ...

### Pre-reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

Parinya Chalermsook, Bundit Laekhanukit, Danupon Nanongkai
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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