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We study the multidimensional vector packing problem with selfish items. An item is a d-dimensional non-zero vector, whose rational components are in [0, 1]. A set of items can be packed into a bin if for every $$1 \le i \le d$$ , the sum of the ith components of all items of this set does not exceed 1. Items share costs of bins proportionally to the $$\ell _1$$ -norms of items, and each item corresponds to a selfish player in the sense that it prefers to be packed into a bin minimizing its resulting cost. This defines a class of games called vector packing games. We show that any game in this class has a packing that is a strong equilibrium, and that both the strong price of anarchy and the strong price of stability are logarithmic in d. We also provide an algorithm that constructs a packing that is a strong equilibrium. Furthermore, we show improved and nearly tight lower and upper bounds of $$d+0.657067$$ and $$d+0.657143$$ , respectively, for any $$d\ge 2$$ , on the price of anarchy. This exhibits a difference between the multidimensional problem and the one-dimensional problem, for which that price of anarchy is at most 1.6428.

The subset sum algorithm is a natural heuristic for the classical Bin Packing problem: In each iteration, the algorithm finds among the unpacked items, a maximum size set of items that fits into a new bin. More than 35 years after its first mention in the literature, establishing the worst-case performance of this heuristic remains, surprisingly, an open problem. Due to their simplicity and intuitive appeal, greedy algorithms are the heuristics of choice of many practitioners. Therefore, better understanding simple greedy heuristics is, in general, an interesting topic in its own right. Very recently, Epstein and Kleiman (Proc. ESA 2008, pp. 368---380) provided another incentive to study the subset sum algorithm by showing that the Strong Price of Anarchy of the game theoretic version of the Bin Packing problem is precisely the approximation ratio of this heuristic. In this paper we establish the exact approximation ratio of the subset sum algorithm, thus settling a long standing open problem. We generalize this result to the parametric variant of the Bin Packing problem where item sizes lie on the interval $$(0, \alpha ]$$(0,?] for some $$\alpha \le 1$$?≤1, yielding tight bounds for the Strong Price of Anarchy for all $$\alpha \le 1$$?≤1. Finally, we study the pure Price of Anarchy of the parametric Bin Packing game for which we show nearly tight upper and lower bounds for all $$\alpha \le 1$$?≤1.

Consider the following scheduling game. A set of jobs, each controlled by a selfish agent, are to be assigned to m uniformly related machines. The cost of a job is defined as the total load of the machine that its job is assigned to. A job is interested in minimizing its cost, while the social objective is maximizing the minimum load (the value of the cover) over the machines. This goal is different from the regular makespan minimization goal, which was extensively studied in a game theoretic context. We study the price of anarchy (poa) and the price of stability (pos) for uniformly related machines. The results are expressed in terms of s, which is the maximum speed ratio between any two machines. For uniformly related machines, we prove that the pos is unbounded for s>2, and the poa is unbounded for s?2. For the remaining cases we show that while the poa grows to infinity as s tends to 2, the pos is at most 2 for any s≤2.

We study the multidimensional vector packing problem with selfish items. An item is d-dimensional non-zero vector, whose rational components are in [0,1], and a set of items can be packed into a bin if for any 1 ≤ i ≤ d, the sum of the ith components of all items of this set does not exceed 1. Items share costs of bins proportionally to their l1-norms, and each item corresponds to a selfish player in the sense that it prefers to be packed into a bin minimizing its resulting cost. This defines a class of games called vector packing games. We show that any game in this class has a packing that is a strong equilibrium, and that the strong price of anarchy (and the strong price of stability) is logarithmic in d, and provide an algorithm that constructs such a packing. We also show improved and nearly tight lower and upper bounds of d + 0.657067 and d + 0.657143 respectively, on the price of anarchy, exhibiting a difference between the multidimensional problem and the one dimensional problem, for which that price of anarchy is at most 1.6428.

We study the multidimensional vector scheduling problem with selfish jobs, both in non-cooperative and in cooperative versions, in two variants. These variants differ in the private costs of the jobs: in the first variant, the cost is the l ∞ norm (maximum over components) while in the second variant it is the l 1 norm (sum of components) of the load. We show existence of assignments that are Nash, strong Nash, weakly and strictly Pareto optimal Nash equilibria in these settings. For the first variant, we improve upon the previous bounds on the price of anarchy for the non-cooperative case, and find tight bounds for every number of machines and dimension. For the cooperative case we provide tight bounds on the strong prices of anarchy and stability, as well as tight bounds on weakly and strictly Pareto optimal prices of anarchy and stability, for every number of machines and dimension. For the second variant, which was not considered before, we again consider all the aforementioned measures, and find tight bounds, each of which being a function of the number of machines and the dimension, showing cardinal differences in the behavior of these measures both between the two variants, and in comparison to the one-dimensional case.

The subset sum algorithm is a natural heuristic for the classical Bin Packing problem: In each iteration, the algorithm finds among the unpacked items, a maximum size set of items that fits into a new bin. More than 35 years after its first mention in the literature, establishing the worst-case performance of this heuristic remains, surprisingly, an open problem. Due to their simplicity and intuitive appeal, greedy algorithms are the heuristics of choice of many practitioners. Therefore, better understanding simple greedy heuristics is, in general, an interesting topic in its own right. Very recently, Epstein and Kleiman (Proc. ESA 2008, pages 368-380) provided another incentive to study the subset sum algorithm by showing that the Strong Price of Anarchy of the game theoretic version of the Bin Packing problem is precisely the approximation ratio of this heuristic. In this paper we establish the exact approximation ratio of the subset sum algorithm, thus settling a long standing open problem. We generalize this result to the parametric variant of the Bin Packing problem where item sizes lie on the interval (0, ?] for some ? ≤ 1, yielding tight bounds for the Strong Price of Anarchy for all ? ≤ 1. Finally, we study the pure Price of Anarchy of the parametric Bin Packing game for which we show nearly tight upper and lower bounds for all ? ≤ 1.

We consider a scheduling problem where each job is controlled by a selfish agent, who is only interested in minimizing its own cost, which is defined as the total load on the machine that its job is assigned to. We consider the objective of maximizing the minimum load (cover) over the machines. Unlike the regular makespan minimization problem, which was extensively studied in a game theoretic context, this problem has not been considered in this setting before. We study the price of anarchy (poa) and the price of stability (pos). We show that on related machines, both these values are unbounded. We then focus on identical machines. We show that the $\textsc{pos}$ is 1, and we derive tight bounds on the $\textsc{poa}$ for m ≤ 6 and nearly tight bounds for general m. In particular, we show that the $\textsc{poa}$ is at least 1.691 for larger m and at most 1.7. Hence, surprisingly, the $\textsc{poa}$ is less than the $\textsc{poa}$ for the makespan problem, which is 2. To achieve the upper bound of 1.7, we make an unusual use of weighting functions. Finally, in contrast we show that the mixed $\textsc{poa}$ grows exponentially with m for this problem, although it is only ?(logm/loglogm) for the makespan.