Bin Packing Games

The bin packing game is a cooperative N -person game, where the set of players consists of k bins, each has capacity 1 and n items of sizes a1, a2, ⋅ ⋅ ⋅ , an, w.l.o.g, we assume 0 ≤ ai ≤ 1 for all 1 ≤ i ≤ n. The value function of a coalition of bins and items is the maximum total size of items in the coalition that can be packed into the bins of the coalition. A typical question of the bin packing game is to study the existence of the core, i.e. given an instance of a bin packing game v, is the core C(v) ∕= ∅ ? If the answer is ‘yes’, then how to find the core allocation of the grand coalition? Instead of directly analyzing the existence of the core, we study by look at the -core, which can be viewed as the generalization of the core because it is the core when = 0. For any instance of the bin packing game, there exists a minimal min such that for all ≥ min, the -core is not empty. The is also called the tax rate, hence the problem becomes to find the minimal tax rate such that the associated -core is nonempty. In chapter 1, we briefly introduce the background of game theory and some concepts from the cooperative game theory. In chapter 2, by studying the fractional bin packing game, we give a sufficient and necessary condition for the existence of the -core and successively summarize some results about the bound of the minimal tax rate. In chapter 3, we study the computational complexity of bin packing games and fractional bin packing games. In chapter 4 and chapter 5, we discuss exact algorithms and approximation algorithms for computing the value function of bin packing games and the corresponding fractional bin packing games, as well as the approximation algorithm for computing the minimal tax rate. Finally, in chapter 6, we summarize the conclusions of previous chapters and further discuss the related unsolved problems we have met. In the end, we present some simple applications of the bin packing game, which are useful in practice.