[Paper Review] A QPTAS for maximum weight independent set of polygons with polylogarithmically many vertices

Anna Adamaszek, Andreas Wiese|arXiv (Cornell University)|Jan 5, 2014

Computational Geometry and Mesh Generation26 references28 citations

TL;DR

This paper presents a quasi-polynomial time approximation scheme (QPTAS) for the Maximum Weight Independent Set of Polygons (MWISP) problem when each polygon has at most polylogarithmically many vertices. It extends a geometric approximation framework by reducing the problem to triangles and introducing a novel plane partitioning method that handles arbitrary polygon angles, achieving a (1 + ε)-approximation in time 2^{poly(log n, 1/ε)}.

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 ne, and it is not even clear whether the problem is APX-hard. We present a (1 + e)-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 2poly(log n, 1/e). In particular, our result implies that for this setting the problem is not APX-hard, unless NP ⊆ DTIME(2poly(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 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 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 framework fail. Also, in general it is not well understood how to handle this difficulty by efficient approximation algorithms.

Motivation & Objective

• To develop an efficient approximation algorithm for the Maximum Weight Independent Set of Polygons (MWISP) problem under the constraint that each polygon has polylogarithmically many vertices.
• To extend a geometric approximation framework—previously effective only for axis-parallel rectangles—to handle arbitrary-angle polygons with arbitrary shapes.
• To overcome the challenge of non-uniform polygon orientations that invalidate prior analysis and partitioning techniques.
• To demonstrate that MWISP with bounded vertex count is not APX-hard unless NP ⊆ DTIME(2^{poly(log n)}), implying strong inapproximability bounds.
• To design a plane partitioning procedure that evenly distributes optimal solution polygons across faces while minimizing edge intersections, even for complex angles.

Proposed method

• Reduce the general MWISP problem to the special case where all input polygons are triangles, preserving the approximation guarantee.
• Adapt the core plane partitioning technique from the original framework to work with arbitrary-angle polygons by introducing a new geometric decomposition method.
• Construct a partition of the plane using few edges such that each polygon from the optimal solution intersects only a constant number of faces and is evenly distributed across them.
• Ensure the new partitioning method maintains the efficiency and approximation properties of the original framework despite the added complexity of non-orthogonal polygons.
• Use dynamic programming on the resulting partitioned faces to compute a (1 + ε)-approximate solution in quasi-polynomial time.
• Prove that the new partitioning procedure avoids the failure modes of prior methods when applied to polygons with arbitrary angles.

Experimental results

Research questions

• RQ1 Can the QPTAS framework for geometric independent set problems be extended beyond axis-parallel rectangles to arbitrary polygons with bounded vertex count?
• RQ2 What modifications are required to maintain the approximation guarantee when input polygons have arbitrary angles and non-rectangular shapes?
• RQ3 Is the MWISP problem with polylogarithmic vertex count APX-hard, or does it admit a QPTAS under plausible complexity assumptions?
• RQ4 Can a plane partitioning method be designed that ensures both low intersection count and even distribution of optimal solution objects across faces for arbitrary polygons?
• RQ5 How does the complexity of handling arbitrary polygon angles affect the analysis and efficiency of existing geometric approximation frameworks?

Key findings

• The paper presents a (1 + ε)-approximation algorithm for MWISP with polylogarithmic vertex count, running in quasi-polynomial time 2^{poly(log n, 1/ε)}.
• The algorithm reduces the general MWISP problem to the case of triangles, enabling the use of specialized geometric techniques.
• A new plane partitioning method is introduced that handles arbitrary polygon angles and ensures each optimal polygon intersects only O(1) faces.
• The framework’s core technical component is adapted to work with non-rectangular polygons without sacrificing approximation quality.
• The result implies that MWISP with bounded vertex count is not APX-hard unless NP ⊆ DTIME(2^{poly(log n)}), a strong complexity-theoretic implication.
• The analysis successfully overcomes obstacles that invalidate prior methods when applied to polygons with arbitrary orientations.