1,528 research outputs found
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]
Hybrid Rounding Techniques for Knapsack Problems
We address the classical knapsack problem and a variant in which an upper
bound is imposed on the number of items that can be selected. We show that
appropriate combinations of rounding techniques yield novel and powerful ways
of rounding. As an application of these techniques, we present a linear-storage
Polynomial Time Approximation Scheme (PTAS) and a Fully Polynomial Time
Approximation Scheme (FPTAS) that compute an approximate solution, of any fixed
accuracy, in linear time. This linear complexity bound gives a substantial
improvement of the best previously known polynomial bounds.Comment: 19 LaTeX page
AFPTAS results for common variants of bin packing: A new method to handle the small items
We consider two well-known natural variants of bin packing, and show that
these packing problems admit asymptotic fully polynomial time approximation
schemes (AFPTAS). In bin packing problems, a set of one-dimensional items of
size at most 1 is to be assigned (packed) to subsets of sum at most 1 (bins).
It has been known for a while that the most basic problem admits an AFPTAS. In
this paper, we develop methods that allow to extend this result to other
variants of bin packing. Specifically, the problems which we study in this
paper, for which we design asymptotic fully polynomial time approximation
schemes, are the following. The first problem is "Bin packing with cardinality
constraints", where a parameter k is given, such that a bin may contain up to k
items. The goal is to minimize the number of bins used. The second problem is
"Bin packing with rejection", where every item has a rejection penalty
associated with it. An item needs to be either packed to a bin or rejected, and
the goal is to minimize the number of used bins plus the total rejection
penalty of unpacked items. This resolves the complexity of two important
variants of the bin packing problem. Our approximation schemes use a novel
method for packing the small items. This new method is the core of the improved
running times of our schemes over the running times of the previous results,
which are only asymptotic polynomial time approximation schemes (APTAS)
An FPTAS for the -modular multidimensional knapsack problem
It is known that there is no EPTAS for the -dimensional knapsack problem
unless . It is true already for the case, when . But, an
FPTAS still can exist for some other particular cases of the problem.
In this note, we show that the -dimensional knapsack problem with a
-modular constraints matrix admits an FPTAS, whose complexity bound
depends on linearly. More precisely, the proposed algorithm complexity
is
where is the linear programming complexity bound. In particular, for
fixed the arithmetical complexity bound becomes Our algorithm is actually a
generalisation of the classical FPTAS for the -dimensional case.
Strictly speaking, the considered problem can be solved by an exact
polynomial-time algorithm, when is fixed and grows as a polynomial
on . This fact can be observed combining previously known results. In this
paper, we give a slightly more accurate analysis to present an exact algorithm
with the complexity bound Note that
the last bound is non-linear by with respect to the given FPTAS
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