text: The Maximum Weight Independent Set of Polygons problem is a fundamental problem in computational geometry. Given a set of weighted polygons in the 2-dimensional plane, the goal is to find a set of pairwise non-overlapping polygons with maximum total weight. Due to its wide range of applications, the MWISP problem and its special cases have been extensively studied both in the approximation algorithms and the computational geometry community. Despite a lot of research, its general case is not well-understood. Currently the best known polynomial time algorithm achieves an approximation ratio of n^(epsilon) [Fox and Pach, SODA 2011], and it is not even clear whether the problem is APX-hard. We present a (1+epsilon)-approximation algorithm, assuming that each polygon in the input has at most a polylogarithmic number of vertices. Our algorithm has quasi-polynomial running time. We use a recently introduced framework for approximating maximum weight independent set in geometric intersection graphs. The framework has been used to construct a QPTAS in the much simpler case of axis-parallel rectangles. We extend it in two ways, to adapt it to our much more general setting. First, we show that its technical core can be reduced to the case when all input polygons are triangles. Secondly, we replace its key technical ingredient which is a method to partition the plane using only few edges such that the objects stemming from the optimal solution are evenly distributed among the resulting faces and each object is intersected only a few times. Our new procedure for this task is not more complex than the original one, and it can handle the arising difficulties due to the arbitrary angles of the polygons. Note that already this obstacle makes the known analysis for the above framework fail. Also, in general it is not well understood how to handle this difficulty by efficient approximation algorithms.
  @cite_16: 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.
  @cite_22: 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.
  @cite_2: A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP. The recognition problem was not known to be decidable until very recently, when two independent papers established exponential upper bounds on the number of intersections needed to realize a string graph [18, 20]. These results implied that the recognition problem lies in NEXP. In the present paper we improve this by showing that the recognition problem for string graphs is in NP, and therefore NP-complete, since Kratochvil [12] showed that the recognition problem is NP-hard. The result has consequences for the computational complexity of problems in graph drawing, and topological inference.
summary: For the general case, the best known result is a @math -approximation algorithm for collections of curves due to Fox and Pach @cite_16 , which assumes that any two curves intersect only @math times, i.e., that they are @math -intersecting. Prior to this, Agarwal and Mustafa presented an algorithm for the special case of straight line segments that computes an independent set of size @math @cite_22 , which yields a worst case approximation ratio of @math . Note that it is @math -hard to decide whether a given graph can be represented as an intersection graph of a collection of curves (i.e., is a string graph) @cite_2 .