Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack

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\u2705]. The best-known polynomial-time approximation factor is O(log log n) [Chalermsook and Chuzhoy, SODA\u2709] and it admits a QPTAS [Adamaszek and Wiese, FOCS\u2713; Chuzhoy and Ene, FOCS\u2716]. 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\u2704]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA\u2715]. 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\u27(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\u2717]

Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments

In the Maximum Weight Independent Set of Rectangles problem (MWISR) we aregiven a weighted set of $n$ axis-parallel rectangles in the plane. The task isto find a subset of pairwise non-overlapping rectangles with the maximumpossible total weight. This problem is NP-hard and the best-knownpolynomial-time approximation algorithm, due to by Chalermsook and Walczak(SODA 2021), achieves approximation factor $O(\log\log n )$. While in theunweighted setting, constant factor approximation algorithms are known, due toMitchell (FOCS 2021) and to G\'alvez et al. (SODA 2022), it remains open toextend these techniques to the weighted setting. In this paper, we consider MWISR through the lens of parameterizedapproximation. Grandoni et al. (ESA 2019) gave a $(1-\epsilon)$-approximationalgorithm with running time $k^{O(k/\epsilon^8)} n^{O(1/\epsilon^8)}$ time,where $k$ is the number of rectangles in an optimum solution. Unfortunately,their algorithm works only in the unweighted setting and they left it as anopen problem to give a parameterized approximation scheme in the weightedsetting. Our contribution is a partial answer to the open question of Grandoni et al.(ESA 2019). We give a parameterized approximation algorithm for MWISR thatgiven a parameter $k$, finds a set of non-overlapping rectangles of weight atleast $(1-\epsilon) \text{opt}_k$ in $2^{O(k \log(k/\epsilon))}n^{O(1/\epsilon)}$ time, where $\text{opt}_k$ is the maximum weight of asolution of cardinality at most $k$. Note that thus, our algorithm may return asolution consisting of more than $k$ rectangles. To complement this apparentweakness, we also propose a parameterized approximation scheme with runningtime $2^{O(k^2 \log(k/\epsilon))} n^{O(1)}$ that finds a solution withcardinality at most $k$ and total weight at least $(1-\epsilon)\text{opt}_k$for the special case of axis-parallel segments.<br

A 3-Approximation Algorithm for Maximum Independent Set of Rectangles

We study the Maximum Independent Set of Rectangles (MISR) problem, where we are given a set of axis-parallel rectangles in the plane and the goal is to select a subset of non-overlapping rectangles of maximum cardinality. In a recent breakthrough, Mitchell [2021] obtained the first constant-factor approximation algorithm for MISR. His algorithm achieves an approximation ratio of 10 and it is based on a dynamic program that intuitively recursively partitions the input plane into special polygons called corner-clipped rectangles, without intersecting certain special horizontal line segments called fences. In this paper, we present a 3-approximation algorithm for MISR which is based on a similar recursive partitioning scheme. First, we use a partition into a more general class of axis-parallel polygons with constant complexity each, which allows us to provide an arguably simpler analysis and at the same time already improves the approximation ratio to 6. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity, we improve our approximation ratio to 3. In particular, our partitioning uses more general fences that can be sequences of up to O(1) line segments each. This and our other new ideas may be useful for future work towards a PTAS for MISR.Comment: 41 page

A fast approximation scheme for the multiple knapsack problem

In this paper we propose an improved efficient approximation scheme for the multiple knapsack problem (MKP). Given a set $A$ of $n$ items and set $B$ of $m$ bins with possibly different capacities, the goal is to find a subset $A'/ \subseteq A$ of maximum total profit that can be packed into $B$ without exceeding the capacities of the bins. Kellerer gave a PTAS for MKP with identical capacities and Chekuri and Khanna presented a PTAS for MKP with arbitrary capacities with running time $n^{O(1/ \epsilon^8 \log(1/\epsilon))}$. Recently we found an EPTAS for MKP with running time $2^{O(1/\epsilon^5 \log(1/\epsilon))} poly(n)$. Here we present an improved EPTAS with running time $2^{O(1/\epsilon \log(1/\epsilon)^4)} poly(n)$. If the modified round-up property for bin packing with different sizes is true, the running time can be improved to $2^{O(1/\epsilon \log(1/\epsilon)^2)} poly(n)$

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum