[Paper Review] Approximation Schemes for Maximum Weight Independent Set of Rectangles

Anna Adamaszek, Andreas Wiese|arXiv (Cornell University)|Jul 6, 2013

Computational Geometry and Mesh Generation4 references14 citations

TL;DR

This paper presents the first quasi-polynomial time $(1+\varepsilon)$-approximation scheme for the Maximum Weight Independent Set of Rectangles (MWISR) problem, using a novel geometric dynamic programming approach that recursively partitions the plane into bounded-complexity polygons. The key contribution is a $(1+\varepsilon)$-approximation in time $2^{\mathrm{poly}(\log n / \varepsilon)}$, resolving a long-standing open problem for rectangles, and implying MWISR is not APX-hard unless NP ⊆ DTIME($2^{\mathrm{polylog}\,n}$).

ABSTRACT

In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to select a maximum weight subset of pairwise non-overlapping rectangles. Due to many applications, e.g. in data mining, map labeling and admission control, the problem has received a lot of attention by various research communities. We present the first (1+epsilon)-approximation algorithm for the MWISR problem with quasi-polynomial running time 2^{poly(log n/epsilon)}. In contrast, the best known polynomial time approximation algorithms for the problem achieve superconstant approximation ratios of O(log log n) (unweighted case) and O(log n / log log n) (weighted case). Key to our results is a new geometric dynamic program which recursively subdivides the plane into polygons of bounded complexity. We provide the technical tools that are needed to analyze its performance. In particular, we present a method of partitioning the plane into small and simple areas such that the rectangles of an optimal solution are intersected in a very controlled manner. Together with a novel application of the weighted planar graph separator theorem due to Arora et al. this allows us to upper bound our approximation ratio by (1+epsilon). Our dynamic program is very general and we believe that it will be useful for other settings. In particular, we show that, when parametrized properly, it provides a polynomial time (1+epsilon)-approximation for the special case of the MWISR problem when each rectangle is relatively large in at least one dimension. Key to this analysis is a method to tile the plane in order to approximately describe the topology of these rectangles in an optimal solution. This technique might be a useful insight to design better polynomial time approximation algorithms or even a PTAS for the MWISR problem.

Motivation & Objective

• Address the long-standing open problem of achieving a $(1+\varepsilon)$-approximation for MWISR in polynomial time.
• Overcome the limitations of existing $O(\log n / \log\log n)$-approximation algorithms for weighted rectangles.
• Develop a general geometric dynamic programming framework applicable to geometric independent set problems with bounded complexity.
• Establish that MWISR is not APX-hard unless NP ⊆ DTIME($2^{\mathrm{polylog}\,n}$), under standard complexity assumptions.

Proposed method

• Propose a geometric dynamic program (GEO-DP) that recursively subdivides the plane into polygons of bounded complexity using a set of lines derived from the optimal solution.
• Use a novel partitioning technique to ensure that rectangles in the optimal solution are intersected only in controlled, low-weight regions.
• Apply the weighted planar graph separator theorem to bound the number of recursive subproblems and control approximation loss.
• Introduce a tiling method for $\delta$-large rectangles (those large in at least one dimension) to enable a polynomial-time $(1+\varepsilon)$-approximation for this special case.
• Leverage random grid offsets to probabilistically minimize the weight of rectangles intersected by grid lines, ensuring low loss in the approximation.
• Parameterize GEO-DP with $k = (1/\varepsilon)(1/\delta)^{O(1)}$ to balance recursion depth and approximation quality.

Experimental results

Research questions

• RQ1 Can a $(1+\varepsilon)$-approximation scheme be designed for MWISR in quasi-polynomial time, despite the lack of constant-factor polynomial-time algorithms?
• RQ2 What geometric and dynamic programming techniques can be used to subdivide the plane such that the optimal solution is preserved up to $\varepsilon$-fractional loss?
• RQ3 Can the structure of $\delta$-large rectangles be exploited to achieve a polynomial-time $(1+\varepsilon)$-approximation?
• RQ4 What is the relationship between the integrality gap of the natural LP relaxation and the complexity of MWISR, and can it be bounded using geometric techniques?
• RQ5 Is MWISR APX-hard, or does it admit a PTAS under plausible complexity assumptions?

Key findings

• The paper presents the first $(1+\varepsilon)$-approximation algorithm for MWISR with quasi-polynomial running time $2^{\mathrm{poly}(\log n / \varepsilon)}$.
• The algorithm achieves an approximation ratio of $1+O(\varepsilon)$ by ensuring that the total weight of rectangles intersected by the partitioning lines is at most $\varepsilon \cdot w(\mathcal{R})$.
• For the special case of $\delta$-large rectangles (where $\max\{g_i, h_i\} \in [K, (1/\delta)K]$), the algorithm yields a polynomial-time $(1+\varepsilon)$-approximation.
• The method implies that MWISR is not APX-hard unless NP ⊆ DTIME($2^{\mathrm{polylog}\,n}$), under standard complexity assumptions.
• The geometric dynamic programming framework is general and may be applicable to other geometric independent set problems.
• The analysis relies on a novel application of the weighted planar graph separator theorem and a random grid offset technique to minimize approximation loss.