Online Bin Covering: Expectations vs. Guarantees

Bin covering is a dual version of classic bin packing. Thus, the goal is to cover as many bins as possible, where covering a bin means packing items of total size at least one in the bin. For online bin covering, competitive analysis fails to distinguish between most algorithms of interest; all "reasonable" algorithms have a competitive ratio of 1/2. Thus, in order to get a better understanding of the combinatorial difficulties in solving this problem, we turn to other performance measures, namely relative worst order, random order, and max/max analysis, as well as analyzing input with restricted or uniformly distributed item sizes. In this way, our study also supplements the ongoing systematic studies of the relative strengths of various performance measures. Two classic algorithms for online bin packing that have natural dual versions are Harmonic and Next-Fit. Even though the algorithms are quite different in nature, the dual versions are not separated by competitive analysis. We make the case that when guarantees are needed, even under restricted input sequences, dual Harmonic is preferable. In addition, we establish quite robust theoretical results showing that if items come from a uniform distribution or even if just the ordering of items is uniformly random, then dual Next-Fit is the right choice.Comment: IMADA-preprint-c

New Results on the Probabilistic Analysis of Online Bin Packing and its Variants

The classical bin packing problem can be stated as follows: We are given a multiset of items {a1, ..., an} with sizes in [0,1], and want to pack them into a minimum number of bins, each of which with capacity one. There are several applications of this problem, for example in the field of logistics: We can interpret the i-th item as time a package deliverer spends for the i-th tour. Package deliverers have a daily restricted working time, and we want to assign the tours such that the number of package deliverers needed is minimized. Another setup is to think of the items as boxes with a standardized basis, but variable height. Then, the goal is to pack these boxes into a container, which is standardized in all three dimensions. Moreover, applications of variants of the classical bin packing problem arise in cloud computing, when we have to store virtual machines on servers. Besides its practical relevance, the bin packing problem is one of the fundamental problems in theoretical computer science: It was proven many years ago that under standard complexity assumptions it is not possible to compute the value of an optimal packing of the items efficiently - classical bin packing is NP-complete. Computing the value efficiently means that the runtime of the algorithm is bounded polynomially in the number of items we have to pack. Besides the offline version, where we know all items at the beginning, also the online version is of interest: Here, the items are revealed one-by-one and have to be packed into a bin immediately and irrevocably without knowing which and how many items will still arrive in the future. Also this version is of practical relevance. In many situations we do not know the whole input at the beginning: For example we are unaware of the requirements of future virtual machines, which have to be stored, or suddenly some more packages have to be delivered, and some deliverers already started their tour. We can think of the classical theoretical analysis of an online algorithm A as follows: An adversary studies the behavior of the algorithm and afterwards constructs a sequence of items I. Then, the performance is measured by the number of used bins by A performing on I, divided by the value of an optimal packing of the items in I. The adversary tries to choose a worst-case sequence so this way to measure the performance is very pessimistic. Moreover, the chosen sequences I often turn out to be artificial: For example, in many cases the sizes of the items increase monotonically over time. Instances in practice are often subject to random influence and therefore it is likely that they are good-natured. In this thesis we analyze the performance of online algorithms with respect to two stochastic models. The first model is the following: The adversary chooses a set of items SI and a distribution F on SI. Then, the items are drawn independently and identically distributed according to F. In the second model the adversary chooses a finite set of items SI and then these items arrive in random order, that is random with respect to the uniform distribution on the set of all possible permutations of the items. It is possible to show that the adversary in the second stochastic model is at least as powerful as in the first one. We can classify the results in this thesis in three parts: In the first part we consider the complexity of classical bin packing and its variants cardinality-constrained and class-constrained bin packing in both stochastic models. That is, we determine if it is possible to construct algorithms that are in expectation nearly optimal for large instances that are constructed according to the stochastic models or if there exist non-trivial lower bounds. Among other things we show that the complexity of class-constrained bin packing differs in the two models under consideration. In the second part we deal with bounded-space bin packing and the dual maximization variant bin covering. We show that it is possible to overcome classical worst-case bounds in both models. In other words, we see that bounded-space algorithms benefit from randomized instances compared to the worst case. Finally, we consider selected heuristics for class-constrained bin packing and the corresponding maximization variant class-constrained bin covering. Here, we note that the different complexity of class-constrained bin packing with respect to the studied stochastic models observed in the first part is not only a theoretical phenomenon, but also takes place for many common algorithmic approaches. Interestingly, when we apply the same algorithmic ideas to class-constrained bin covering, we benefit from both types of randomization similarly. </ul

Probabilistic analysis of algorithms for dual bin packing problems

In the dual bin packing problem, the objective is to assign items of given size to the largest possible number of bins, subject to the constraint that the total size of the items assigned to any bin is at least equal to 1. We carry out a probabilistic analysis of this problem under the assumption that the items are drawn independently from the uniform distribution on [0, 1] and reveal the connection between this problem and the classical bin packing problem as well as to renewal theory.

TS2PACK: A Two-Level Tabu Search for the Three-dimensional Bin Packing Problem

Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing one

Online Bin Covering with Limited Migration

Semi-online models where decisions may be revoked in a limited way have been studied extensively in the last years. This is motivated by the fact that the pure online model is often too restrictive to model real-world applications, where some changes might be allowed. A well-studied measure of the amount of decisions that can be revoked is the migration factor beta: When an object o of size s(o) arrives, the decisions for objects of total size at most beta * s(o) may be revoked. Usually beta should be a constant. This means that a small object only leads to small changes. This measure has been successfully investigated for different, classical problems such as bin packing or makespan minimization. The dual of makespan minimization - the Santa Claus or machine covering problem - has also been studied, whereas the dual of bin packing - the bin covering problem - has not been looked at from such a perspective. In this work, we extensively study the bin covering problem with migration in different scenarios. We develop algorithms both for the static case - where only insertions are allowed - and for the dynamic case, where items may also depart. We also develop lower bounds for these scenarios both for amortized migration and for worst-case migration showing that our algorithms have nearly optimal migration factor and asymptotic competitive ratio (up to an arbitrary small epsilon). We therefore resolve the competitiveness of the bin covering problem with migration

Extreme-Point-based Heuristics for the Three-Dimensional Bin Packing problem

One of the main issues in addressing three-dimensional packing problems is finding an efficient and accurate definition of the points at which to place the items inside the bins, because the performance of exact and heuristic solution methods is actually strongly influenced by the choice of a placement rule. We introduce the extreme point concept and present a new extreme point-based rule for packing items inside a three-dimensional container. The extreme point rule is independent from the particular packing problem addressed and can handle additional constraints, such as fixing the position of the items. The new extreme point rule is also used to derive new constructive heuristics for the three-dimensional bin-packing problem. Extensive computational results show the effectiveness of the new heuristics compared to state-of-the-art results. Moreover, the same heuristics, when applied to the two-dimensional bin-packing problem, outperform those specifically designed for the proble