7 research outputs found
The convergence time for selfish bin packing
In classic bin packing, the objective is to partition a set of n items with positive rational sizes in (0, 1] into a minimum number of subsets called bins, such that the total size of the items of each bin at most 1. We study a bin packing game where the cost of each bin is 1, and given a valid packing of the items, each item has a cost associated with it, such that the items that are packed into a bin share its cost equally. We find tight bounds on the exact worst-case number of steps in processes of convergence to pure Nash equilibria. Those are processes that are given an arbitrary packing as an initial packing. As long as there exists an item that can reduce its cost by moving from its bin to another bin, in each step, a controller selects such an item and instructs it to perform such a beneficial move. The process converges when no further beneficial moves exist. The tight function of n that we find is in Θ(n 3/2 ). This improves the previous bound of Ma et al. [14], who showed an upper bound of O(n 2)
Generalized selfish bin packing
Standard bin packing is the problem of partitioning a set of items with
positive sizes no larger than 1 into a minimum number of subsets (called bins)
each having a total size of at most 1. In bin packing games, an item has a
positive weight, and given a valid packing or partition of the items, each item
has a cost or a payoff associated with it. We study a class of bin packing
games where the payoff of an item is the ratio between its weight and the total
weight of items packed with it, that is, the cost sharing is based linearly on
the weights of items. We study several types of pure Nash equilibria: standard
Nash equilibria, strong equilibria, strictly Pareto optimal equilibria, and
weakly Pareto optimal equilibria. We show that any game of this class admits
all these types of equilibria. We study the (asymptotic) prices of anarchy and
stability (PoA and PoS) of the problem with respect to these four types of
equilibria, for the two cases of general weights and of unit weights. We show
that while the case of general weights is strongly related to the well-known
First Fit algorithm, and all the four PoA values are equal to 1.7, this is not
true for unit weights. In particular, we show that all of them are strictly
below 1.7, the strong PoA is equal to approximately 1.691 (another well-known
number in bin packing) while the strictly Pareto optimal PoA is much lower. We
show that all the PoS values are equal to 1, except for those of strong
equilibria, which is equal to 1.7 for general weights, and to approximately
1.611824 for unit weights. This last value is not known to be the (asymptotic)
approximation ratio of any well-known algorithm for bin packing. Finally, we
study convergence to equilibria
Packing, Scheduling and Covering Problems in a Game-Theoretic Perspective
Many packing, scheduling and covering problems that were previously
considered by computer science literature in the context of various
transportation and production problems, appear also suitable for describing and
modeling various fundamental aspects in networks optimization such as routing,
resource allocation, congestion control, etc. Various combinatorial problems
were already studied from the game theoretic standpoint, and we attempt to
complement to this body of research.
Specifically, we consider the bin packing problem both in the classic and
parametric versions, the job scheduling problem and the machine covering
problem in various machine models. We suggest new interpretations of such
problems in the context of modern networks and study these problems from a game
theoretic perspective by modeling them as games, and then concerning various
game theoretic concepts in these games by combining tools from game theory and
the traditional combinatorial optimization. In the framework of this research
we introduce and study models that were not considered before, and also improve
upon previously known results.Comment: PhD thesi