66 research outputs found
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)
Asymptotic results for the Generalized Bin Packing Problem
We present a worst case analysis for the Generalized Bin Packing Problem, a novel packing problem arising in many Transportation and Logistics settings and characterized by multiple item and bin attributes and by the joint presence of both compulsory and non-compulsory items. As a preliminary worst case analysis has recently been proposed in the literature, we extend this study by proposing semi-online and offline algorithms, extending the well known First Fit Decreasing and Best Fit Decreasing heuristics for the Bin Packing Problem. In particular, we show that knowing part of the instance or the whole instance is not enough for computing worst case ratio bounds
Vector Bin Packing with Multiple-Choice
We consider a variant of bin packing called multiple-choice vector bin
packing. In this problem we are given a set of items, where each item can be
selected in one of several -dimensional incarnations. We are also given
bin types, each with its own cost and -dimensional size. Our goal is to pack
the items in a set of bins of minimum overall cost. The problem is motivated by
scheduling in networks with guaranteed quality of service (QoS), but due to its
general formulation it has many other applications as well. We present an
approximation algorithm that is guaranteed to produce a solution whose cost is
about times the optimum. For the running time to be polynomial we
require and . This extends previous results for vector
bin packing, in which each item has a single incarnation and there is only one
bin type. To obtain our result we also present a PTAS for the multiple-choice
version of multidimensional knapsack, where we are given only one bin and the
goal is to pack a maximum weight set of (incarnations of) items in that bin
Algorithms for the Bin Packing Problem with Scenarios
This paper presents theoretical and practical results for the bin packing
problem with scenarios, a generalization of the classical bin packing problem
which considers the presence of uncertain scenarios, of which only one is
realized. For this problem, we propose an absolute approximation algorithm
whose ratio is bounded by the square root of the number of scenarios times the
approximation ratio for an algorithm for the vector bin packing problem. We
also show how an asymptotic polynomial-time approximation scheme is derived
when the number of scenarios is constant. As a practical study of the problem,
we present a branch-and-price algorithm to solve an exponential model and a
variable neighborhood search heuristic. To speed up the convergence of the
exact algorithm, we also consider lower bounds based on dual feasible
functions. Results of these algorithms show the competence of the
branch-and-price in obtaining optimal solutions for about 59% of the instances
considered, while the combined heuristic and branch-and-price optimally solved
62% of the instances considered
On the generalized bin packing problem
The generalized bin packing problem (GBPP) is a novel packing problem arising in many transportation and logistic settings, characterized by multiple items and bins attributes and the presence of both compulsory and non-compulsory items. In this paper, we study the computational complexity and the approximability of the GBPP. We prove that the GBPP cannot be approximated by any constant, unless P = NP. We also study the particular case of a single bin type and show that when an unlimited number of bins is available, the GBPP can be reduced to the bin packing with rejection (BPR) problem, which is approximable. We also prove that the GBPP satisfies Bellman’s optimality principle and, exploiting this result, we develop a dynamic programming solution approach. Finally, we study the behavior of standard and widespread heuristics such as the first fit, best fit, first fit decreasing, and best fit decreasing.We show that while they successfully approximate previous versions of bin packing problems, they fail to approximate the GBPP
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