290 research outputs found
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
Parametric Polyhedra with at least Lattice Points: Their Semigroup Structure and the k-Frobenius Problem
Given an integral matrix , the well-studied affine semigroup
\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be
stratified by the number of lattice points inside the parametric polyhedra
. Such families of parametric polyhedra appear in
many areas of combinatorics, convex geometry, algebra and number theory. The
key themes of this paper are: (1) A structure theory that characterizes
precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{
Sg}(A) such that has at least solutions. We
demonstrate that this set is finitely generated, it is a union of translated
copies of a semigroup which can be computed explicitly via Hilbert bases
computations. Related results can be derived for those right-hand-side vectors
for which has exactly solutions or fewer
than solutions. (2) A computational complexity theory. We show that, when
, are fixed natural numbers, one can compute in polynomial time an
encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function,
using a short sum of rational functions. As a consequence, one can identify all
right-hand-side vectors of bounded norm that have at least solutions. (3)
Applications and computation for the -Frobenius numbers. Using Generating
functions we prove that for fixed the -Frobenius number can be
computed in polynomial time. This generalizes a well-known result for by
R. Kannan. Using some adaptation of dynamic programming we show some practical
computations of -Frobenius numbers and their relatives
Non-projectability of polytope skeleta
We investigate necessary conditions for the existence of projections of
polytopes that preserve full k-skeleta. More precisely, given the combinatorics
of a polytope and the dimension e of the target space, what are obstructions to
the existence of a geometric realization of a polytope with the given
combinatorial type such that a linear projection to e-space strictly preserves
the k-skeleton. Building on the work of Sanyal (2009), we develop a general
framework to calculate obstructions to the existence of such realizations using
topological combinatorics. Our obstructions take the form of graph colorings
and linear integer programs. We focus on polytopes of product type and
calculate the obstructions for products of polygons, products of simplices, and
wedge products of polytopes. Our results show the limitations of constructions
for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge
product surfaces of R\"orig & Ziegler (2009) and complement their results.Comment: 18 pages, 2 figure
Rotational placement of irregular polygons over containers with fixed dimensions using simulated annealing and no-fit polygons
This work deals with the problem of minimizing the waste of space that occurs on a rotational placement of a set of irregular bi-dimensional small items inside a bi-dimensional large object. This problem is approached with an heuristic based on simulated annealing. Traditional " external penalization" techniques are avoided through the application of the no-fit polygon, that determinates the collision-free region for each small item before its placement. The simulated annealing controls: the rotation applied and the placement of the small item. For each non-placed small item, a limited depth binary search is performed to find a scale factor that when applied to the small item, would allow it to be fitted in the large object. Three possibilities to define the sequence on which the small items are placed are studied: larger-first, random permutation and weight sorted. The proposed algorithm is suited for non-convex small items and large objects
Approximating Smallest Containers for Packing Three-dimensional Convex Objects
We investigate the problem of computing a minimal-volume container for the
non-overlapping packing of a given set of three-dimensional convex objects.
Already the simplest versions of the problem are NP-hard so that we cannot
expect to find exact polynomial time algorithms. We give constant ratio
approximation algorithms for packing axis-parallel (rectangular) cuboids under
translation into an axis-parallel (rectangular) cuboid as container, for
cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary
convex container, and for packing convex polyhedra under rigid motions into an
axis-parallel cuboid or arbitrary convex container. This work gives the first
approximability results for the computation of minimal volume containers for
the objects described
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