Bin Covering Research Articles

There is no APTAS for 2-dimensional vector bin packing: Revisited

We study the Vector Bin Packing and the Vector Bin Covering problems, multidimensional generalizations of the Bin Packing and the Bin Covering problems, respectively. In the Vector Bin Packing, we are given a set of d-dimensional vectors from [0,1]d and the aim is to partition the set into the minimum number of bins such that for each bin B, each component of the sum of the vectors in B is at most 1. Woeginger [17] claimed that the problem has no APTAS for dimensions greater than or equal to 2. We note that there was a slight oversight in the original proof. In this work, we give a revised proof using some additional ideas from [3,8]. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than 600599.An instance of Vector Bin Packing is called δ-skewed if every item has at most one dimension greater than δ. As a natural extension of our general d-Dimensional Vector Bin Packing result we show that for ε∈(0,12500) it is NP-hard to obtain a (1+ε)-approximation for δ-Skewed Vector Bin Packing if δ>20ε.In the Vector Bin Covering problem given a set of d-dimensional vectors from [0,1]d, the aim is to obtain a family of disjoint subsets (called bins) with the maximum cardinality such that for each bin B, each component of the sum of the vectors in B is at least 1. Using ideas similar to our Vector Bin Packing result, we show that for Vector Bin Covering there is no APTAS for dimensions greater than or equal to 2. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than 998997.

The bin covering with delivery problem, extended investigations for the online case

We deal with a model that is called Bin Covering with Delivery. Here we search for a “good and fast” covering. This problem has been defined and investigated recently. Here we continue and significantly augment the preliminary investigations of the problem. After studying the goodness of adapted versions of some classical algorithms, we propose a parametrized algorithm and use a heuristic parameter optimization method to solve this (algorithmically very challenging) problem. Namely, we apply local search to determine a good choice of the parameters. The efficiency of the proposed method is demonstrated by intensive computer experiments on appropriate modifications of benchmark instances.

On the Use of Equivalence Classes for Optimal and Suboptimal Bin Packing and Bin Covering

Bin packing and bin covering are important optimization problems in many industrial fields, such as packaging, recycling, and food processing. The problem concerns a set of items, each with its own value, that are to be sorted into bins in such a way that the total value of each bin, as measured by the sum of its item values, is not above (for packing) or below (for covering) a given target value. The optimization problem concerns minimizing, for bin packing, or maximizing, for bin covering, the number of bins. This is a combinatorial NP-hard problem, for which true optimal solutions can only be calculated in specific cases, such as when restricted to a small number of items. To get around this problem, many suboptimal approaches exist. This article describes the formulations of the bin packing and covering problems that allow finding the true optimum, for instance, counting hundreds of items using general-purpose MILP-solvers. Also presented are suboptimal solutions that come within less than 10% of the optimum while taking significantly less time to calculate, even ten to 100 times faster, depending on the required accuracy.

Online Bin Covering with Advice

The bin covering problem asks for covering a maximum number of bins with an online sequence of $n$ items of different sizes in the range $(0,1]$; a bin is said to be covered if it receives items of total size at least 1. We study this problem in the advice setting and provide tight bounds for the size of advice required to achieve optimal solutions. Moreover, we show that any algorithm with advice of size $o(\log \log n)$ has a competitive ratio of at most 0.5. In other words, advice of size $o(\log \log n)$ is useless for improving the competitive ratio of 0.5, attainable by an online algorithm without advice. This result highlights a difference between the bin covering and the bin packing problems in the advice model: for the bin packing problem, there are several algorithms with advice of constant size that outperform online algorithms without advice. Furthermore, we show that advice of size $O(\log \log n)$ is sufficient to achieve a competitive ratio that is arbitrarily close to $0.53\bar{3}$ and hence strictly better than the best ratio $0.5$ attainable by purely online algorithms. The technicalities involved in introducing and analyzing this algorithm are quite different from the existing results for the bin packing problem and confirm the different nature of these two problems. Finally, we show that a linear number of bits of advice is necessary to achieve any competitive ratio better than 15/16 for the online bin covering problem.

On Lazy Bin Covering and Packing problems

In this paper, we study two interesting variants of the classical bin packing problem, called Lazy Bin Covering (LBC) and Cardinality Constrained Maximum Resource Bin Packing (CCMRBP) problems. For the offline LBC problem, we first prove the approximation ratio of the First-Fit-Decreasing and First-Fit-Increasing algorithms, then present an APTAS. For the online LBC problem, we give a competitive analysis for the algorithms of Next-Fit, Worst-Fit, First-Fit, and a modified HARMONIC M algorithm. The CCMRBP problem is a generalization of the Maximum Resource Bin Packing (MRBP) problem Boyar et al. (2006) [1]. For this problem, we prove that its offline version is no harder to approximate than the offline MRBP problem.

Class Constrained Bin Covering

We study the following variant of the bin covering problem. We are given a set of unit sized items, where each item has a color associated with it. We are given an integer parameter k≥1 and an integer bin size B≥k. The goal is to assign the items (or a subset of the items) into a maximum number of subsets of at least B items each, such that in each such subset the total number of distinct colors of items is at least k. We study both the offline and the online variants of this problem. We first design an optimal polynomial time algorithm for the offline problem. For the online problem we give a lower bound of 1+H k−1 (where H k−1 denotes the (k−1)-th harmonic number), and an O(k)-competitive algorithm. Finally, we analyze the performance of the natural heuristic First fit.

Improved Approximation Algorithms for Maximum Resource Bin Packing and Lazy Bin Covering Problems

In this paper, we study two variants of the bin packing and covering problems called Maximum Resource Bin Packing (MRBP) and Lazy Bin Covering (LBC) problems, and present new approximation algorithms for them. For the offline MRBP problem, the previous best known approximation ratio is $\frac{6}{5}$(=1.2) achieved by the classical First-Fit-Increasing (FFI) algorithm (Boyar et al. in Theor. Comput. Sci. 362(1–3):127–139, 2006). In this paper, we give a new FFI-type algorithm with an approximation ratio of $\frac{80}{71}$(≈1.12676). For the offline LBC problem, it has been shown in Lin et al. (COCOON, pp. 340–349, 2006) that the classical First-Fit-Decreasing (FFD) algorithm achieves an approximation ratio of $\frac{71}{60}$(≈1.18333). In this paper, we present a new FFD-type algorithm with an approximation ratio of $\frac{17}{15}$(≈1.13333). Our algorithms are based on a pattern-based technique and a number of other observations. They run in near linear time (i.e., O(nlog n)), and therefore are practical.

Online LIB problems: Heuristics for Bin Covering and lower bounds for Bin Packing

We consider the NP Hard problems of online Bin Covering and Packing while requiring that larger (or longer, in the one dimensional case) items be placed at the bottom of the bins, below smaller (or shorter) items — we call such a version, the LIB version of problems. Bin sizes can be uniform or variable. We look at computational studies for both the Best Fit and Harmonic Fit algorithms for uniform sized bin covering. The Best Fit heuristic for this version of the problem is introduced here. The approximation ratios obtained were well within the theoretical upper bounds. For variable sized bin covering, a more thorough analysis revealed definite trends in the maximum and average approximation ratios. Finally, we prove that for online LIB bin packing with uniform size bins, no heuristic can guarantee an approximation ratio better than 1.76 under the online model considered.

A Note on Optimal Bin Packing and Optimal Bin Covering with Items of Random Size

Consider a probability measure $\mu$ on $[0, 1]$ and independent identically distributed random variables $X_1,\cdots, X_n$ distributed according to $\mu$. Denote by $Q_n = Q_n (X_1,\cdots, X_n)$ the minimum number of unit-size bins needed to pack items of size $X_1,\cdots, X_n$. Previous estimates of $Q_n$ are considerably improved and simplified. Similar estimates are obtained for the maximum number of unit-size bins that can be covered by $X_1,\cdots, X_n$.

Optimal Bin Covering with Items of Random Size

Consider a probability measure $\mu $ on $[0,1]$ and independent random variables $X_1 , \cdots ,X_n $ distributed according to $\mu $. Let $Q_n = Q_n (X_1 , \cdots ,X_n )$ be the largest number of unit-size bins that can be covered by items of size $X_1 , \cdots ,X_n $. The properties of $Q_n $ are investigated in the spirit of our work on optimal bin packing with items of random sizes. While the covering problem has many similarities with the packing problem, it turns out to be significantly harder.