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

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

Online Bin Covering with Exact Parameter Advice

We show an asymptotic 2/3-competitive strategy for the bin covering problem using O(b+log n) bits of advice, where b is the number of bits used to encode a rational value and n is the length of the input sequence.Comment: 10 pages, 2 figure, submitted to Informatic

A Simple PTAS for the Dual Bin Packing Problem and Advice Complexity of Its Online Version

Recently, Renault (2016) studied the dual bin packing problem in the per-request advice model of online algorithms. He showed that given O(1/eps) advice bits for each input item allows approximating the dual bin packing problem online to within a factor of 1+eps. Renault asked about the advice complexity of dual bin packing in the tape-advice model of online algorithms. We make progress on this question. Let s be the maximum bit size of an input item weight. We present a conceptually simple online algorithm that with total advice O((s + log n)/eps^2) approximates the dual bin packing to within a 1+eps factor. To this end, we describe and analyze a simple offline PTAS for the dual bin packing problem. Although a PTAS for a more general problem was known prior to our work (Kellerer 1999, Chekuri and Khanna 2006), our PTAS is arguably simpler to state and analyze. As a result, we could easily adapt our PTAS to obtain the advice-complexity result. We also consider whether the dependence on s is necessary in our algorithm. We show that if s is unrestricted then for small enough eps > 0 obtaining a 1+eps approximation to the dual bin packing requires Omega_eps(n) bits of advice. To establish this lower bound we analyze an online reduction that preserves the advice complexity and approximation ratio from the binary separation problem due to Boyar et al. (2016). We define two natural advice complexity classes that capture the distinction similar to the Turing machine world distinction between pseudo polynomial time algorithms and polynomial time algorithms. Our results on the dual bin packing problem imply the separation of the two classes in the advice complexity world

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