[Paper Review] Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack

Fabrizio Grandoni, Stefan Kratsch+1 more|arXiv (Cornell University)|Jan 1, 2019

Computational Geometry and Mesh Generation31 references3 citations

TL;DR

This paper presents parameterized approximation schemes (PAS) for two geometric packing problems: maximum independent set of rectangles (MISR) and 2D geometric knapsack with rotations (2DKR). It achieves a PAS for MISR running in time $k^{O(k/ϵ^8)}n^{O(1/ϵ^8)}$ and for 2DKR with similar efficiency, offering $(1+\varepsilon)$-approximate solutions in FPT-like time for fixed $\varepsilon$, overcoming W[1]-hardness barriers through grid decomposition and kernelization techniques.

ABSTRACT

The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+epsilon)-approximations in f(k,epsilon)n^O(1) time where k is some parameter of the input. The goal is to overcome lower bounds from either of the areas. We obtain the following results on parameterized approximability: - In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(log log n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant epsilon>0 and integer k>0, in time f(k,epsilon)n^g(epsilon), either outputs a solution of size at least k/(1+epsilon), or declares that the optimum solution has size less than k. - In the (2-dimensional) geometric knapsack problem (2DK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of 2DK with rotations (2DKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factor is 2+epsilon [Jansen and Zhang, SODA'04]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for 2DKR. For all considered problems, getting time f(k,epsilon)n^O(1), rather than f(k,epsilon)n^g(epsilon), would give FPT time f'(k)n^O(1) exact algorithms by setting epsilon=1/(k+1), contradicting W[1]-hardness. Instead, for each fixed epsilon>0, our PASs give (1+epsilon)-approximate solutions in FPT time. For both MISR and 2DKR our techniques also give rise to preprocessing algorithms that take n^g(epsilon) time and return a subset of at most k^g(epsilon) rectangles/items that contains a solution of size at least k/(1+epsilon) if a solution of size k exists. This is a special case of the recently introduced notion of a polynomial-size approximate kernelization scheme [Lokshtanov et al., STOC'17].

Motivation & Objective

• . The paper aims to develop parameterized approximation schemes (PAS) for NP-hard geometric packing problems to overcome intractability barriers.
• . It specifically targets the maximum independent set of rectangles (MISR) and 2D geometric knapsack with rotations (2DKR), which are W[1]-hard when parameterized by solution size k.
• . The goal is to design algorithms that return a $(1+\varepsilon)$-approximate solution in time $f(k,\varepsilon)n^{O(1)}$ for any fixed $\varepsilon > 0$, avoiding FPT time that would contradict W[1]-hardness.
• . It also introduces a polynomial-size approximate kernelization scheme (PSAKS) for MISR, reducing the input to $k^{O(1/\varepsilon^8)}$ rectangles while preserving a $(1+\varepsilon)$-approximate solution.
• . The work addresses the challenge of extending such schemes to the weighted versions and to 2DKR without rotations, which remain open problems.

Proposed method

• . Uses a $k \times k$ grid construction to ensure each input rectangle contains a grid crossing point, enabling decomposition into independent subproblems.
• . Applies a non-trivial adaptation of Frederickson’s planar graph decomposition to partition the grid into disjoint cell groups, each containing a constant number of optimal solution rectangles.
• . Solves each subproblem in $n^{O(1/\varepsilon^8)}$ time via dynamic programming or enumeration, leveraging the bounded number of optimal rectangles per group.
• . Employs a kernelization technique that reduces the input to $k^{O(1/\varepsilon^8)}$ rectangles while preserving a $k/(1+\varepsilon)$-approximate solution, forming a PSAKS.
• . For 2DKR, uses a similar strategy: partitions the knapsack into regions using width/height constraints and applies a recursive decomposition with auxiliary items to enforce tight packing bounds.
• . Proves W[1]-hardness for both 2DK and 2DKR via a parameterized reduction from multi-subset sum, showing that FPT algorithms would imply FPT algorithms for W[1]-hard problems.

Experimental results

Research questions

• RQ1 . Can a parameterized approximation scheme (PAS) be designed for the maximum independent set of rectangles (MISR) problem, achieving a $(1+\varepsilon)$-approximation in time $f(k,\varepsilon)n^{O(1)}$?
• RQ2 . Is it possible to design a PAS for the 2D geometric knapsack problem with rotations (2DKR), even though the problem is W[1]-hard?
• RQ3 . Can the kernelization approach be extended to preserve a $(1+\varepsilon)$-approximate solution in polynomial time while reducing the input size to $k^{O(1/\varepsilon^8)}$ rectangles for MISR?
• RQ4 . Does the proposed method generalize to the weighted versions of MISR and 2DKR, where rectangles have associated profits?
• RQ5 . Can a PAS be constructed for 2DK without rotations, given that the current approach relies on rotation to enforce structural constraints?

Key findings

• . A parameterized approximation scheme (PAS) for MISR is achieved with a running time of $k^{O(k/\varepsilon^8)}n^{O(1/\varepsilon^8)}$, providing a $(1+\varepsilon)$-approximate solution.
• . The paper presents a polynomial-size approximate kernelization scheme (PSAKS) for MISR, reducing the input to $k^{O(1/\varepsilon^8)}$ rectangles while preserving a solution of size at least $k/(1+\varepsilon)$.
• . A PAS for 2DKR is developed with similar time complexity, achieving a $(1+\varepsilon)$-approximation in $f(k,\varepsilon)n^{O(1)}$ time.
• . The paper proves that both 2DK and 2DKR are W[1]-hard when parameterized by solution size $k$, ruling out FPT algorithms unless W[1] = FPT.
• . The kernelization technique ensures that any $c$-approximate solution on the reduced instance yields a $c(1+\varepsilon)$-approximate solution for the original instance.
• . The approach relies on grid decomposition and auxiliary items to enforce tight packing constraints, enabling the use of dynamic programming on small subproblems.