Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack

TL;DR

This paper develops parameterized approximation schemes for maximum independent set of rectangles and geometric knapsack problems, providing fixed-parameter tractable algorithms that approximate solutions within a factor of (1+eps).

Contribution

It introduces the first parameterized approximation schemes for MISR and TDKR, advancing the understanding of fixed-parameter approximability for these geometric problems.

Findings

Provides a PAS for MISR with time f(k,eps)n^{g(eps)}.

Shows W[1]-hardness for TDK and TDKR.

Offers a PAS for TDKR, the first for this variant.

Abstract

The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+eps)-approximations in f(k,eps)n^{O(1)} time where k is some parameter of the input. We obtain the following results on parameterized approximability: 1) 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(loglog 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 eps>0 and integer k>0, in time f(k,eps)n^{g(eps)}, either outputs a solution of size at least k/(1+eps), or declares that the optimum solution has size less than k. 2) In the (2-dimensional) geometric knapsack problem (TDK) 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 TDK with rotations (TDKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factors are 558/325+eps and 4/3+eps, resp. [Galvez et al., FOCS'17]. 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 TDKR. For all considered problems, getting time f(k,eps)n^{O(1)}, rather than f(k,eps)n^{g(eps)}, would give FPT time f'(k)n^{O(1)} exact algorithms using eps=1/(k+1), contradicting W[1]-hardness.

Full text

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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 $\mathsf{W[1]}$-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 $\varepsilon>0$ and integer $k>0$, in time $f(k,\varepsilon)n^{g(\varepsilon)}$, either outputs a solution of size at least $k/(1+\varepsilon)$, or declares that the optimum solution has size less than $k$.

• $\bullet$

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 $2+\varepsilon$ [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 $\mathsf{W[1]}$-hard. Furthermore, we present a PAS for 2dkr.

For all considered problems, getting time $f(k,\varepsilon)n^{O(1)}$, rather than $f(k,\varepsilon)n^{g(\varepsilon)}$, would give FPT time $f^{\prime}(k)n^{O(1)}$ exact algorithms by setting $\varepsilon=1/(k+1)$, contradicting $\mathsf{W[1]}$-hardness. Instead, for each fixed $\varepsilon>0$, our PASs give $(1+\varepsilon)$-approximate solutions in FPT time.

For both misr and 2dkr our techniques also give rise to preprocessing algorithms that take $n^{g(\varepsilon)}$ time and return a subset of at most $k^{g(\varepsilon)}$ rectangles/items that contains a solution of size at least $k/(1+\varepsilon)$ 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].

1 Introduction

Approximation algorithms and parameterized algorithms are two well-established ways to deal with NP-hard problems. An $\alpha$-approximation for an optimization problem is a polynomial-time algorithm that computes a feasible solution whose cost is within a factor $\alpha$ (that might be a function of the input size $n$) of the optimal cost. In particular, a polynomial-time approximation scheme (PTAS) is a $(1+\varepsilon)$-approximation algorithm running in time $n^{g(\varepsilon)}$, where $\varepsilon>0$ is a given constant and $g$ is some computable function. In parameterized algorithms we identify a parameter $k$ of the input, that we informally assume to be much smaller than $n$. The goal here is to solve the problem optimally in fixed-parameter tractable (FPT) time $f(k)n^{O(1)}$, where $f$ is some computable function. Recently, researchers started to combine the two notions (see, e.g., the survey by Marx [Marx08]). The idea is to design approximation algorithms that run in FPT (rather than polynomial) time, e.g., to get $(1+\varepsilon)$-approximate solutions in time $f(k,\varepsilon)n^{O(1)}$. In this paper we continue this line of research on parameterized approximation, and apply it to two fundamental rectangle packing problems.

1.1 Our results and techniques

Our focus is on parameterized approximation algorithms. Unfortunately, as observed by Marx [Marx08], when the parameter $k$ is the desired solution size, computing $(1+\varepsilon)$-approximate solutions in time $f(k,\varepsilon)n^{O(1)}$ implies fixed-parameter tractability. Indeed, setting $\varepsilon=1/(k+1)$ guarantees to find an optimal solution when that value equals to $k\in\mathbb{N}$ and we get time $f(k,1/(k+1))n^{O(1)}=f^{\prime}(k)n^{O(1)}$. Since the considered problems are $\mathsf{W[1]}$-hard (in part, this is established in our work), they are unlikely to be FPT and similarly unlikely to have such nice approximation schemes.

Instead, we construct algorithms (for two maximization problems) that, given $\varepsilon>0$ and an integer $k$, take time $f(k,\varepsilon)n^{g(\varepsilon)}$ and either return a solution of size at least $k/(1+\varepsilon)$ or declare that the optimum is less than $k$. We call such an algorithm a parameterized approximation scheme (PAS). Note that if we run such an algorithm for each $k^{\prime}\leq k$ then we can guarantee that we compute a solution with cardinality at least $\min\{k,\mathrm{OPT}\}/(1+\varepsilon)$ where $\mathrm{OPT}$ denotes the size of the optimal solution. So intuitively, for each $\varepsilon>0$, we have an FPT-algorithm for getting a $(1+\varepsilon)$-approximate solution.

In this paper we consider the following two geometric packing problems, and design PASs for them.

Maximum Independent Set of Rectangles.

In the maximum independent set of rectangles problem (misr) we are given a set of $n$ axis-parallel rectangles $\mathcal{R}=\{R_{1},\dots,R_{n}\}$ in the two-dimensional plane, where $R_{i}$ is the open set of points $(x_{i}^{(1)},x_{i}^{(2)})\times(y_{i}^{(1)},y_{i}^{(2)})$. A feasible solution is a subset of rectangles $\mathcal{R}^{\prime}\subseteq\mathcal{R}$ such that for any two rectangles $R,R^{\prime}\in\mathcal{R}^{\prime}$ we have $R\cap R^{\prime}=\emptyset$. Our objective is to find a feasible solution of maximum cardinality $|\mathcal{R}^{\prime}|$. W.l.o.g. we assume that $x_{i}^{(1)},y_{i}^{(1)},x_{i}^{(2)},y_{i}^{(2)}\in\{0,\dots,2n-1\}$ for each $R_{i}\in\mathcal{R}$ (see e.g. [adamaszek2013approximation]).

misr is very well-studied in the area of approximation algorithms. The problem is known to be NP-hard [fowler1981optimal], and the current best polynomial-time approximation factor is $O(\log\log n)$ for the cardinality case [CC2009] (addressed in this paper), and $O(\log n/\log\log n)$ for the natural generalization with rectangle weights [chan2012approximation]. The cardinality case also admits a $(1+\varepsilon)$-approximation with a running time of $n^{poly(\log\log(n/\varepsilon))}$ [chuzhoy2016approximating] and there is a (slower) QPTAS known for the weighted case [adamaszek2013approximation]. The problem is also known to be $\mathsf{W[1]}$-hard w.r.t. the number $k$ of rectangles in the solution [Marx05], and thus unlikely to be solvable in FPT time $f(k)n^{O(1)}$.

In this paper we achieve the following main result:

Theorem 1.1**.**

There is a PAS for

In order to achieve the above result, we combine several ideas. Our starting point is a polynomial-time construction of a $k\times k$ grid such that each rectangle in the input contains some crossing point of this grid (or we find a solution of size $k$ directly). By applying (in a non-trivial way) a result by Frederickson [federickson1987fast] on planar graphs, and losing a small factor in the approximation, we define a decomposition of our grid into a collection of disjoint groups of cells. Each such group defines an independent instance of the problem, consisting of the rectangles strictly contained in the considered group of cells. Furthermore, we guarantee that each group spans only a constant number $O_{\varepsilon}(1)$ of rectangles of the optimum solution. Therefore in FPT time we can guess the correct decomposition, and solve each corresponding subproblem in $n^{O_{\varepsilon}(1)}$ time. We remark that our approach deviates substantially from prior work, and might be useful for other related problems.

An adaptation of our construction also leads to the following $(1+\epsilon)$-approximative kernelization.

Theorem 1.3**.**

There is an algorithm for

Similarly as for a PAS, if we run the above algorithm for each $k^{\prime}\leq k$ we obtain a set of size $k^{O(1/\epsilon^{8})}$ that contains a solution of size at least $\min\{k,\mathrm{OPT}\}/(1+\varepsilon)$. Observe that any $c$-approximate solution on the obtained set of rectangles is also a feasible, and $c(1+\varepsilon)$-approximate, solution for the original instance if $\mathrm{OPT}\leq k$ and otherwise has size at least $k/(c(1+\varepsilon))$. Thus, our result is a special case of a polynomial-size approximate kernelization scheme (PSAKS) as defined in [LokshtanovPRS17].111The definition due to Lokshtanov et al. [LokshtanovPRS17] is not restricted to generating a small subset of the input and a dedicated solution lifting algorithm may be used.

2-Dimensional Geometric Knapsack.

In the (2-Dimensional) Geometric Knapsack problem () we are given a square knapsack $[0,N]\times[0,N]$, $N\in\mathbb{N}$, and a set of $n$ items $I$, where each item $i\in I$ is an open rectangle $(0,w_{i})\times(0,h_{i})$, $N\geq w_{i},h_{i}\in\mathbb{N}$. The goal is to find a feasible packing of a subset $I^{\prime}\subseteq I$ of the items of maximum cardinality $|I^{\prime}|$. Such packing maps each item $i\in I^{\prime}$ into a new translated rectangle $(a_{i},a_{i}+w_{i})\times(b_{i},b_{i}+h_{i})$222Intuitively, $i$ is shifted by $a_{i}$ to the right and by $b_{i}$ to the top., so that the translated rectangles are fully contained in the knapsack and do not overlap with each other. Here we also consider a variant of

Problem 1.6**.**

2DKwith rotations* (*