A QPTAS for Maximum Weight Independent Set of Polygons with Polylogarithmically Many Vertices

Abstract

The Maximum Weight Independent Set of Polygons (MWISP) problem is a fundamental problem in computational geometry. Given a set of weighted polygons in the two-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 and connections to other problems, 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 yet. Currently the best known polynomial time algorithm achieves an approximation ratio of n , and it is not even clear whether the problem is APX-hard. We present a (1 + )-approximation algorithm, assuming that each polygon in the input has at most a polylogarithmic number of vertices. Our algorithm has quasi-polynomial running time, i.e., it runs in time 2 poly(log n,1/ ) . In particular, our result implies that for this setting the problem is not APX-hard, unless NP ⊆ DTIME(2 poly(log n) ). 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 axisparallel 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 no more complicated than the original one and, importantly, it can handle the difficulties arising from the arbitrary angles of the input polygons. Note that already this obstacle makes the known analysis for the above *

Independent Set of Convex Polygons: From $$n^{\epsilon }$$ n ϵ to $$1+\epsilon $$ 1 + ϵ

“…Going beyond polynomial time results, for independent set of arbitrary polygons there is a QPTAS [3,17], i.e., a (1 + )-approximation in time n (log n) O (1) , building on an earlier QPTAS for axis-parallel rectangles [2]. This implies that all the above problems are not APX-hard, unless NP ⊆ DTIME(n poly(log n) ).…”

“…Our algorithm is a geometric divide-and-conquer algorithm similar to the algorithm used in [1,3]. It recursively divides the area containing the input polygons into smaller and smaller pieces.…”

“…Similarly as before, assume that there is a grid line 3 of G i such that 3 intersects the boundary of P 1 at a point that lies on the boundary of Aū j . We compute a path Q though Aū j as above.…”

“…On the other hand, the best complexity result shows only strong NP-hardness [14,19] which leaves an enormous gap. Even more, there is a QPTAS [3,17] which suggests that much better polynomial time approximation results are possible.…”

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

“…Baker's approach was extended by Eppstein [Epp00] to graphs with bounded local treewidth, and by Grohe [Gro03] to graphs excluding minors. Separators have also played a key role in geometric optimization algorithms, including: (i) PTAS for independent set and (continuous) piercing set for fat objects [Cha03,MR10], (ii) QPTAS for maximum weighted independent sets of polygons [AW13,AW14,Har14], and (iii) QPTAS for Set Cover by pseudodisks [MRR14a], among others. Lastly, Cabello and Gajser [CG14a] develop PTAS's for some of the problems we study in the specific setting of minor-free graphs.…”