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.

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

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

Two new Probability inequalities and Concentration Results

Concentration results and probabilistic analysis for combinatorial problems like the TSP, MWST, graph coloring have received much attention, but generally, for i.i.d. samples (i.i.d. points in the unit square for the TSP, for example). Here, we prove two probability inequalities which generalize and strengthen Martingale inequalities. The inequalities provide the tools to deal with more general heavy-tailed and inhomogeneous distributions for combinatorial problems. We prove a wide range of applications - in addition to the TSP, MWST, graph coloring, we also prove more general results than known previously for concentration in bin-packing, sub-graph counts, Johnson-Lindenstrauss random projection theorem. It is hoped that the strength of the inequalities will serve many more purposes.Comment: 3

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

Selfish Bin Covering

In this paper, we address the selfish bin covering problem, which is greatly related both to the bin covering problem, and to the weighted majority game. What we mainly concern is how much the lack of coordination harms the social welfare. Besides the standard PoA and PoS, which are based on Nash equilibrium, we also take into account the strong Nash equilibrium, and several other new equilibria. For each equilibrium, the corresponding PoA and PoS are given, and the problems of computing an arbitrary equilibrium, as well as approximating the best one, are also considered.Comment: 16 page