  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.
