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On Approximating Maximum Independent Set of Rectangles [article]

Julia Chuzhoy, Alina Ene
2016 arXiv   pre-print
We study the Maximum Independent Set of Rectangles (MISR) problem: given a set of n axis-parallel rectangles, find a largest-cardinality subset of the rectangles, such that no two of them overlap.  ...  We introduce several new technical ideas, that we hope will lead to further progress on this and related problems.  ...  Introduction In the Maximum Independent Set of Rectangles (MISR) problem, the input is a set R of n axis-parallel rectangles, and the goal is to find a maximum-cardinality subset of the rectangles, such  ... 
arXiv:1608.00271v1 fatcat:474ivplvjfemlfr52u4ggpdpnq

Approximation Schemes for Maximum Weight Independent Set of Rectangles

Anna Adamaszek, Andreas Wiese
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to select a maximum weight subset of pairwise non-overlapping  ...  one dimension.  ...  Related Work The maximum weight independent set of rectangles problem has been widely studied.  ... 
doi:10.1109/focs.2013.50 dblp:conf/focs/AdamaszekW13 fatcat:rydxfcr3czd5xmxlekmlaaxwry

Label placement by maximum independent set in rectangles

Pankaj K. Agarwal, Marc van Kreveld, Subhash Suri
1998 Computational geometry  
In O(n log n) time, we can find an O(log n)factor approximation of the maximum subset in a set of n arbitrary axis-parallel rectangles in the plane.  ...  Motivated by the problem of labeling maps, we investigate the problem of computing a large non-intersecting subset in a set of n rectangles in the plane. Our results are as follows.  ...  Compute 112, the (real) maximum independent set of R12. Recursively compute I1 and 12, the approximate maximum independent sets in R1 and R2, respectively. 4.  ... 
doi:10.1016/s0925-7721(98)00028-5 fatcat:5asvvnxbnre2nhmny32uusch7y

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.  ...  This work is in part supported by VR grant 621-2002-4049 1 Recall that minimum vertex cover is always complement of maximum independent set.  ...  intersection graphs of orthogonal rectangles in the plane (e.g., the problem of maximum independent set on box graphs is known to admit an O(log n)-approximation polynomial-time algorithm [1]).  ... 
doi:10.1007/3-540-45071-8_7 fatcat:hkrtpfpjdjdklhcpkqflivlyq4

Matching colored points with rectangles

L. E. Caraballo, C. Ochoa, P. Pérez-Lantero, J. Rojas-Ledesma
2015 Journal of combinatorial optimization  
The approximation results are based on a rela-18 tion of our problem with the problem of finding a maximum independent 19 set in a family of axis-aligned rectangles.  ...  Find a maximum bichromatic matching of S with rectangles. 9 For each problem we provide a polynomial-time approximation algorithm 10 that constructs a matching with at least 1/4 of the number of rectan  ...  Let I 1 be a (1/2)-approximation for the maximum independent 229 set in R 1 and I 2 be a (1/2)-approximation for the maximum independent set 230 in R 2 (Lemma 3).  ... 
doi:10.1007/s10878-015-9971-x fatcat:spce3ca3jzanpe53qfc6gtbp2e

Matching colored points with rectangles [article]

L. E. Caraballo, C. Ochoa, P. Pérez-Lantero, J. Rojas-Ledesma
2014 arXiv   pre-print
The approximation results are based on a relation of our problem with the problem of finding a maximum independent set in a family of axis-aligned rectangles.  ...  Find a maximum monochromatic matching of S with rectangles. 2. Find a maximum bichromatic matching of S with rectangles.  ...  Let I 1 be a (1/2)-approximation for the maximum independent set in R 1 and I 2 be a (1/2)-approximation for the maximum independent set in R 2 (Lemma 3).  ... 
arXiv:1309.3696v2 fatcat:mfwldohiajfqtiisfo5u752juq

Greedy is good: An experimental study on minimum clique cover and maximum independent set problems for randomly generated rectangles [article]

Ritankar Mandal and Anirban Ghosh and Sasanka Roy and Subhas C. Nandy
2012 arXiv   pre-print
Given a set R={R_1,R_2,..., R_n} of n randomly positioned axis parallel rectangles in 2D, the problem of computing the minimum clique cover (MCC) and maximum independent set (MIS) for the intersection  ...  Finally we will provide refined greedy algorithms based on a concept of simplicial rectangle.  ...  the performance of our proposed two greedy heuristics M IS and M IS I for the maximum independent set problem on rectangle intersection graph.  ... 
arXiv:1212.0640v1 fatcat:eccdsw3ryvhn7f2ugn3obw4mcq

Independent Set of Intersection Graphs of Convex Objects in 2D [chapter]

Pankaj K. Agarwal, Nabil H. Mustafa
2004 Lecture Notes in Computer Science  
We present approximation algorithms for computing a maximum independent set of intersection graphs of convex objects in R 2 .  ...  (α/(2 log(2n/α))) 1/3 in time O(n 3 + τ (S)), assuming that S can be preprocessed in time τ (S) to answer certain primitive operations on these convex sets, and (iii) a set of n rectangles with maximum  ...  Since at most one rectangle from the independent set can be in a clique, each iteration removes at most one rectangle from the optimal independent set of S, hence α(S j ) α(S) − j .  ... 
doi:10.1007/978-3-540-27810-8_12 fatcat:dl4xz2tcbzdvbjsqz4wyejw7a4

Independent set of intersection graphs of convex objects in 2D

Pankaj K. Agarwal, Nabil H. Mustafa
2006 Computational geometry  
We present approximation algorithms for computing a maximum independent set of intersection graphs of convex objects in R 2 .  ...  (α/(2 log(2n/α))) 1/3 in time O(n 3 + τ (S)), assuming that S can be preprocessed in time τ (S) to answer certain primitive operations on these convex sets, and (iii) a set of n rectangles with maximum  ...  Since at most one rectangle from the independent set can be in a clique, each iteration removes at most one rectangle from the optimal independent set of S, hence α(S j ) α(S) − j .  ... 
doi:10.1016/j.comgeo.2005.12.001 fatcat:asbydjotmffwpkn7mhxhrh5o3a

Maximum Independent Set of Rectangles [chapter]

Parinya Chalermsook, Julia Chuzhoy
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms  
We study the Maximum Independent Set of Rectangles (MISR) problem: given a collection R of n axisparallel rectangles, find a maximum-cardinality subset of disjoint rectangles.  ...  MISR is a special case of the classical Maximum Independent Set problem, where the input is restricted to intersection graphs of axisparallel rectangles.  ...  Maximum Independent Set is probably one of the most fundamental and extensively studied combinatorial optimization problems.  ... 
doi:10.1137/1.9781611973068.97 fatcat:s7h2e2mufnc2dfqkbjw24bht5y

A note on maximum independent sets in rectangle intersection graphs

Timothy M. Chan
2004 Information Processing Letters  
Finding the maximum independent set in the intersection graph of n axis-parallel rectangles is NP-hard. We re-examine two known approximation results for this problem.  ...  similar algorithm running in only O(n log n + n∆ k−1 ) time, where ∆ ≤ n denotes the maximum number of rectangles a point can be in.  ...  , or equivalently, find a maximum independent set in the intersection graph of the rectangles.  ... 
doi:10.1016/j.ipl.2003.09.019 fatcat:oeg6ftox3rb2nlmqfuk7wqmyti

On Packing Almost Half of a Square with Anchored Rectangles: A Constructive Approach [article]

Sandip Banerjee, Aritra Banik, Bhargab B. Bhattacharya, Arijit Bishnu,, Soumyottam Chatterjee
2014 arXiv   pre-print
The puzzle has been popularized of late by Peter Winkler Winkler2007. Let P_n be a set of n points, including the origin, in the unit square U = [0,1]^2.  ...  We would term the above rectangles as anchored rectangles. The longstanding conjecture has been that at least half of U can be covered when anchored rectangles are properly placed.  ...  Weighted maximum independent set of axis parallel rectangles In the problem of Maximum Weight Independent Set of Rectangles (MWISR), the input is a set of n axisparallel rectangles R = {R 1 , . . . , R  ... 
arXiv:1401.0108v4 fatcat:6gdeiz72bvexlnowjaav3lm4bi

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  ...  Finally, we use axis-aligned rectangles to illustrate that the same method may be used to derive linear-time approximations for problems on other geometric intersection graph classes.  ...  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

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  ...  Finally, we use axis-aligned rectangles to illustrate that the same method may be used to derive linear-time approximations for problems on other geometric intersection graph classes.  ...  An extended abstract of this paper appeared in the 12th Workshop on Approximation and Online Algorithms (WAOA 2014).  ... 
doi:10.1142/s0218195917500078 fatcat:jckw4etsjrbpnd37ixjbf4iime

Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line: Algorithms and Complexity

José Correa, Laurent Feuilloley, Pablo Pérez-Lantero, José A. Soto
2015 Discrete & Computational Geometry  
Given a set of n axis-parallel rectangles in the plane, finding a maximum independent set (MIS), a maximum weighted independent set (WMIS), and a minimum hitting set (MHS), are basic problems in computational  ...  An interesting case is when there exists a diagonal line that intersects each of the given rectangles.  ...  We also thank two anonymous reviewers whose many suggestions greatly improved the presentation of the paper.  ... 
doi:10.1007/s00454-014-9661-y fatcat:ztwmds5ygnbqlpjgmjzxfy3hne
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