Amir Epstein

Fast convergence to nearly optimal solutions in potential games

We study the speed of convergence of decentralized dynamics to approximately optimal solutions in potential games. We consider α-Nash dynamics in which a player makes a move if the improvement in his payoff is more than an α factor of his own payoff. Despite the known polynomial convergence of α-Nash dynamics to approximate Nash equilibria in symmetric congestion games [7], it has been shown that the convergence time to approximate Nash equilibria in asymmetric congestion games is exponential [25]. In contrast to this negative result, and as the main result of this paper, we show that for asymmetric congestion games with linear and polynomial delay functions, the convergence time of α-Nash dynamics to an approximate optimal solution is polynomial in the number of players, with approximation ratio that is arbitrarily close to the price of anarchy of the game. In particular, we show this polynomial convergence under the minimal liveness assumption that each player gets at least one chance to move in every T steps. We also prove that the same polynomial convergence result does not hold for (exact) best-response dynamics, showing the α-Nash dynamics is required. We extend these results for congestion games to other potential games including weighted congestion games with linear delay functions, cut games (also called party affiliation games) and market sharing games.

Strong equilibrium in cost sharing connection games

We study network games in which each player wishes to connect his source and sink, and the cost of each edge is shared among its users either equally (in Fair Connection Games--FCGs) or arbitrarily (in General Connection Games--GCGs). We study the existence and quality of strong equilibria (SE)--strategy profiles from which no coalition can improve the cost of each of its members--in these settings. We show that SE always exist in the following games: (1) Single source and sink FCGs and GCGs. (2) Single source multiple sinks FCGs and GCGs on series parallel graphs. (3) Multi source and sink FCGs on extension parallel graphs. As for the quality of the SE, in any FCG with n players, the cost of any SE is bounded by H(n) (i.e., the harmonic sum), contrasted with the [Theta](n) price of anarchy. For any GCG, any SE is optimal.

Convex programming for scheduling unrelated parallel machines

We consider the classical problem of scheduling parallel unrelated machines. Each job is to be processed by exactly one machine. Processing job j on machine i requires time pij. The goal is to find a schedule that minimizes the lp norm. Previous work showed a 2-approximation algorithm for the problem with respect to the l∞ norm. For any fixed lp norm the previously known approximation algorithm has a performance of θ(p). We provide a 2-approximation algorithm for any fixed lp norm (p>1). This algorithm uses convex programming relaxation. We also give a √ 2-approximation algorithm for the l2 norm. This algorithm relies on convex quadratic programming relaxation. To the best of our knowledge, this is the first time that general convex programming techniques (apart from SDPs and CQPs) are used in the area of scheduling. We show for any given lp norm a PTAS for any fixed number of machines. We also consider the multidimensional generalization of the problem in which the jobs are d-dimensional. Here the goal is to minimize the lp norm of the generalized load vector, which is a matrix where the rows represent the machines and the columns represent the jobs dimension. For this problem we give a (d+1)-approximation algorithm for any fixed lp norm (p>1).

The hardness of network design for unsplittable flow with selfish users

In this paper we consider the network design for selfish users problem, where we assume the more realistic unsplittable model in which the users can have general demands and each user must choose a single path between its source and its destination. This model is also called atomic (weighted) network congestion game. The problem can be presented as follows : given a network, which edges should be removed to minimize the cost of the worst Nash equilibrium? We consider both computational issues and existential issues (i.e. the power of network design). We give inapproximability results and approximation algorithms for this network design problem. For networks with linear edge latency functions we prove that there is no approximation algorithm for this problem with approximation ratio less then

Load balancing of temporary tasks in the l p norm

(3+\sqrt{5})/2 \approx 2.618

Load Balancing of Temporary Tasks in the ℓ p Norm

unless P=NP. We also show that for networks with polynomials of degree d edge latency functions there is no approximation algorithm for this problem with approximation ratio less then

The Price of Routing Unsplittable Flow

d^{{\it \Theta}(d)}

Efficient graph topologies in network routing games

unless P=NP. Moreover, we observe that the trivial algorithm that builds the entire network is optimal for linear edge latency functions and has an approximation ratio of

A quasi-PTAS for unsplittable flow on line graphs

d^{{\it \Theta}(d)}