Lower bounds for several online variants of bin packing

We consider several previously studied online variants of bin packing and prove new and improved lower bounds on the asymptotic competitive ratios for them. For that, we use a method of fully adaptive constructions. In particular, we improve the lower bound for the asymptotic competitive ratio of online square packing significantly, raising it from roughly 1.68 to above 1.75.Comment: WAOA 201

New bounds for multi-dimensional packing

New upper and lower bounds are presented for a multi-dimensional generalization of bin packing called box packing. Several variants of this problem, including bounded space box packing, square packing, variable sized box packing and resource augmented box packing are also studied. The main results, stated for d=2, are as follows: A new upper bound of 2.66013 for online box packing, a new $14/9 + varepsilon$ polynomial time offline approximation algorithm for square packing, a new upper bound of 2.43828 for online square packing, a new lower bound of 1.62176 for online square packing, a new lower bound of 2.28229 for bounded space online square packing and a new upper bound of 2.32571 for online two-sized box packing

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

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

A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game

We prove a tight lower bound on the asymptotic performance ratio $\rho$ of the bounded space online $d$-hypercube bin packing problem, solving an open question raised in 2005. In the classic $d$-hypercube bin packing problem, we are given a sequence of $d$-dimensional hypercubes and we have an unlimited number of bins, each of which is a $d$-dimensional unit hypercube. The goal is to pack (orthogonally) the given hypercubes into the minimum possible number of bins, in such a way that no two hypercubes in the same bin overlap. The bounded space online $d$-hypercube bin packing problem is a variant of the $d$-hypercube bin packing problem, in which the hypercubes arrive online and each one must be packed in an open bin without the knowledge of the next hypercubes. Moreover, at each moment, only a constant number of open bins are allowed (whenever a new bin is used, it is considered open, and it remains so until it is considered closed, in which case, it is not allowed to accept new hypercubes). Epstein and van Stee [SIAM J. Comput. 35 (2005), no. 2, 431-448] showed that $\rho$ is $\Omega(\log d)$ and $O(d/\log d)$, and conjectured that it is $\Theta(\log d)$. We show that $\rho$ is in fact $\Theta(d/\log d)$. To obtain this result, we elaborate on some ideas presented by those authors, and go one step further showing how to obtain better (offline) packings of certain special instances for which one knows how many bins any bounded space algorithm has to use. Our main contribution establishes the existence of such packings, for large enough $d$, using probabilistic arguments. Such packings also lead to lower bounds for the prices of anarchy of the selfish $d$-hypercube bin packing game. We present a lower bound of $\Omega(d/\log d)$ for the pure price of anarchy of this game, and we also give a lower bound of $\Omega(\log d)$ for its strong price of anarchy

Locality-preserving allocations Problems and coloured Bin Packing

We study the following problem, introduced by Chung et al. in 2006. We are given, online or offline, a set of coloured items of different sizes, and wish to pack them into bins of equal size so that we use few bins in total (at most $\alpha$ times optimal), and that the items of each colour span few bins (at most $\beta$ times optimal). We call such allocations $(\alpha, \beta)$-approximate. As usual in bin packing problems, we allow additive constants and consider $(\alpha,\beta)$ as the asymptotic performance ratios. We prove that for \eps>0, if we desire small $\alpha$, no scheme can beat (1+\eps, \Omega(1/\eps))-approximate allocations and similarly as we desire small $\beta$, no scheme can beat (1.69103, 1+\eps)-approximate allocations. We give offline schemes that come very close to achieving these lower bounds. For the online case, we prove that no scheme can even achieve $(O(1),O(1))$-approximate allocations. However, a small restriction on item sizes permits a simple online scheme that computes (2+\eps, 1.7)-approximate allocations