We consider the Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families @math where @math is connected for every pair of rectangles @math . This algorithm extends to intersection graphs of pseudo-disks. The second algorithm achieves a factor of @math in general rectangle families, for any fixed @math , and works also for the weighted variant of the problem. Both algorithms exploit the plane properties of axis-parallel rectangles in a non-trivial way.
, and its dual counterpart , have been previously studied in many geometric intersection graphs other than rectangle graphs. Gavril @cite_16 gave a polynomial-time algorithm for both of these problem in chordal graphs, intersection graphs of subgraphs of a tree. Apostolico @cite_20 gave a polynomial-time algorithm for these two problems in intersection graphs of chords on a circle, which were later improved by Cenek and Stuart @cite_28 , while Golumbic and Hammer @cite_0 gave a polynomial-time algorithm for intersection graphs of arcs on a circle which was later improved in @cite_9 . A good survey of many generalizations of these results can be found in @cite_46 @cite_17 . Hochbaum and Maass and later Chleb ' k and Chleb ' i kov ' a considered intersection graphs of @math -dimensional boxes in @math @cite_33 @cite_26 , while Erlebach @cite_5 considered intersection graphs of general fat objects in the plane. @cite_4 @cite_8 , approximation algorithms were suggested for and in the class of multiple-interval graphs.
@cite_26 @cite_4 @cite_33 @cite_8 @cite_28 @cite_9 @cite_0 @cite_5 @cite_46 @cite_16 @cite_20 @cite_17
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The intersection graph of a set of geometric objects is defined as a graph G = (S, E) in which there is an edge between two nodes si, sj ∈ S if si ∩ sj ≠ 0. The problem of computing a maximum independent set in the intersection graph of a set of objects is known to be NP-complete for most cases in two and higher dimensions. We present approximation algorithms for computing a maximum independent set of intersection graphs of convex objects in R2. Specifically, given (i) a set of n line segments in the plane with maximum independent set of size α, we present algorithms that find an independent set of size at least (α (2 log(2n α)))1 2 in time O(n3) and (α (2log(2n α)))1 4 in time O(n4 3 logcn), (ii) a set of n convex objects with maximum independent set of size α, we present an algorithm that finds an independent set of size at least (α (2log(2n α)))1 3 in time O(n3 + 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 independent set of size βn, for β≤ 1, we present an algorithm that computes an independent set of size Ω (β2n). All our algorithms use the notion of partial orders that exploit the geometric structure of the convex objects. In this paper, we show that many graph search problems can be solved quite efficiently for a geometric intersection graph of horizontal and vertical line segments. We first extract several basic operations for depth first search and breadth first search on a graph. Then we present data structures for the intersection graph in terms of which those operations can be implemented in an efficient manner. The data structures enable us to solve various graph search problems besides depth first search and breadth first search. Specifying the results obtained in this paper for an intersection graph of n horizontal and vertical segments with m pairs of intersecting segments, we obtain algorithms with the following complexity, where @math . (i) Depth first search and breadth first search can be executed in @math time and @math space. (ii) The biconnected components can be found in @math time and @math space. (iii) A maximum matching and a maximum independent set can be found in $O... Abstract In this paper we continue the study of the edge intersection graphs of single bend paths on a rectangular grid (i.e., the edge intersection graphs where each vertex is represented by one of the following shapes: ⌞ , ⌜ , ⌟ , ⌝ ). These graphs, called B 1 - EPG graphs, were first introduced by (2009) [Golumbic, M. C., M. Lipshteyn and M. Stern, Edge intersection graphs of single bend paths on a grid , Networks 54 :3 (2009), 130–138]. We focus on the class [⌞] (the edge intersection graphs of ⌞-shapes) and show that testing for membership in [⌞] is NP-complete. We then give a characterization and polytime recognition algorithm for special subclasses of Split ∩ [ ⌞ ] . We also consider the natural subclasses of B 1 -EPG formed by the subsets of the four single bend shapes (i.e., ⌞ , ⌞ , ⌜ , ⌞ , ⌝ , ⌞ , ⌜ , ⌝ – note: all other subsets are isomorphic to these up to 90 degree rotation). We observe the expected strict inclusions and incomparability (i.e., [ ⌞ ] ⊊ [ ⌞ , ⌜ ] , [ ⌞ , ⌝ ] ⊊ [ ⌞ , ⌜ , ⌝ ] ⊊ B 1 -EPG and [ ⌞ , ⌜ ] is incomparable with [ ⌞ , ⌝ ] ). Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the independence number, α(GC), of the intersection graph GC of C, obtained by connecting two elements of C with an edge if and only if their intersection is nonempty. This is known to be an NP-hard task even for systems of segments in the plane with at most two different slopes. The best known polynomial time approximation algorithm for systems of arbitrary segments is due to Agarwal and Mustafa, and returns in the worst case an n1 2+o(1)-approximation for α. Using extensions of the Lipton-Tarjan separator theorem, we improve this result and present, for every e > 0, a polynomial time algorithm for computing α(GC) with approximation ratio at most ne. In contrast, for general graphs, for any e > 0 it is NP-hard to approximate the independence number within a factor of n1−e. We also give a subexponential time exact algorithm for computing the independence number of intersection graphs of arcwise connected sets in the plane.