401 research outputs found
Approximating Geometric Knapsack via L-packings
We study the two-dimensional geometric knapsack problem (2DK) in which we are
given a set of n axis-aligned rectangular items, each one with an associated
profit, and an axis-aligned square knapsack. The goal is to find a
(non-overlapping) packing of a maximum profit subset of items inside the
knapsack (without rotating items). The best-known polynomial-time approximation
factor for this problem (even just in the cardinality case) is (2 + \epsilon)
[Jansen and Zhang, SODA 2004].
In this paper, we break the 2 approximation barrier, achieving a
polynomial-time (17/9 + \epsilon) < 1.89 approximation, which improves to
(558/325 + \epsilon) < 1.72 in the cardinality case. Essentially all prior work
on 2DK approximation packs items inside a constant number of rectangular
containers, where items inside each container are packed using a simple greedy
strategy. We deviate for the first time from this setting: we show that there
exists a large profit solution where items are packed inside a constant number
of containers plus one L-shaped region at the boundary of the knapsack which
contains items that are high and narrow and items that are wide and thin. As a
second major and the main algorithmic contribution of this paper, we present a
PTAS for this case. We believe that this will turn out to be useful in future
work in geometric packing problems.
We also consider the variant of the problem with rotations (2DKR), where
items can be rotated by 90 degrees. Also, in this case, the best-known
polynomial-time approximation factor (even for the cardinality case) is (2 +
\epsilon) [Jansen and Zhang, SODA 2004]. Exploiting part of the machinery
developed for 2DK plus a few additional ideas, we obtain a polynomial-time (3/2
+ \epsilon)-approximation for 2DKR, which improves to (4/3 + \epsilon) in the
cardinality case.Comment: 64pages, full version of FOCS 2017 pape
On the Two-Dimensional Knapsack Problem for Convex Polygons
We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly into the knapsack. We allow to rotate the polygons by arbitrary angles. We present a quasi-polynomial time O(1)-approximation algorithm for the general case and a polynomial time O(1)-approximation algorithm if all input polygons are triangles, both assuming polynomially bounded integral input data. Also, we give a quasi-polynomial time algorithm that computes a solution of optimal weight under resource augmentation, i.e., we allow to increase the size of the knapsack by a factor of 1+? for some ? > 0 but compare ourselves with the optimal solution for the original knapsack. To the best of our knowledge, these are the first results for two-dimensional geometric knapsack in which the input objects are more general than axis-parallel rectangles or circles and in which the input polygons can be rotated by arbitrary angles
Smoothing the gap between NP and ER
We study algorithmic problems that belong to the complexity class of the
existential theory of the reals (ER). A problem is ER-complete if it is as hard
as the problem ETR and if it can be written as an ETR formula. Traditionally,
these problems are studied in the real RAM, a model of computation that assumes
that the storage and comparison of real-valued numbers can be done in constant
space and time, with infinite precision. The complexity class ER is often
called a real RAM analogue of NP, since the problem ETR can be viewed as the
real-valued variant of SAT.
In this paper we prove a real RAM analogue to the Cook-Levin theorem which
shows that ER membership is equivalent to having a verification algorithm that
runs in polynomial-time on a real RAM. This gives an easy proof of
ER-membership, as verification algorithms on a real RAM are much more versatile
than ETR-formulas. We use this result to construct a framework to study
ER-complete problems under smoothed analysis. We show that for a wide class of
ER-complete problems, its witness can be represented with logarithmic
input-precision by using smoothed analysis on its real RAM verification
algorithm. This shows in a formal way that the boundary between NP and ER
(formed by inputs whose solution witness needs high input-precision) consists
of contrived input. We apply our framework to well-studied ER-complete
recognition problems which have the exponential bit phenomenon such as the
recognition of realizable order types or the Steinitz problem in fixed
dimension.Comment: 31 pages, 11 figures, FOCS 2020, SICOMP 202
Logic based Benders' decomposition for orthogonal stock cutting problems
We consider the problem of packing a set of rectangular items into a strip of fixed width, without overlapping, using minimum height. Items must be packed with their edges parallel to those of the strip, but rotation by 90\ub0 is allowed. The problem is usually solved through branch-and-bound algorithms. We propose an alternative method, based on Benders' decomposition. The master problem is solved through a new ILP model based on the arc flow formulation, while constraint programming is used to solve the slave problem. The resulting method is hybridized with a state-of-the-art branch-and-bound algorithm. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. We additionally show that the algorithm can be successfully used to solve relevant related problems, like rectangle packing and pallet loading
Approximation algorithms for 2d packing problems
In this thesis we address such 2-dimensional packing problems as strip packing, bin packing and storage packing. These problems play an important role in many application areas, e.g. cutting stock, VLSI design, image processing, and multiprocessor scheduling. The larger part of work is devoted to the storage packing problem, that is the problem of packing weighted rectangles into a single rectangle so as to maximize the total weight of the packed rectangles. Despite the practical importance of the problem, there are just few known results in the literature. The main objective was to fill this gap and also to build the bridges to already known algorithmic solutions for strip packing and bin packing problems. This was successfully achieved. Considering natural relaxations of the storage packing problem we proposed a number of efficient algorithms which are able to find solutions within a factor of (1-\epsilon) of the optimum in polynomial time
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