Elena Kleiman

Parametric Packing of Selfish Items and the Subset Sum Algorithm

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.

Maximizing the Minimum Load: The Cost of Selfishness

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

Selfish Bin Packing

\textsc{pos}

Resource augmented semi-online bounded space bin packing

is 1, and we derive tight bounds on the

Selfish Vector Packing

\textsc{poa}

Scheduling selfish jobs on multidimensional parallel machines

for m ≤ 6 and nearly tight bounds for general m. In particular, we show that the

On the quality and complexity of pareto equilibria in the job scheduling game

\textsc{poa}

The cost of selfishness for maximizing the minimum load on uniformly related machines

is at least 1.691 for larger m and at most 1.7. Hence, surprisingly, the

Packing, Scheduling and Covering Problems in a Game-Theoretic Perspective

\textsc{poa}