A logarithmic approximation for polymatroid congestion games

Abstract

We study the problem of computing a social optimum (minimum cost solution) in polymatroid congestion games, where the strategy space of every player consists of the set of vectors in a playerspecific integral polymatroid base polyhedron defined on the ground set of resources. For general non-decreasing cost functions we devise an H rk -approximation algorithm, where rk is the sum of the ranks of the player-specific polymatroids and H rk denotes the rk-th harmonic number. The main idea of our algorithm is to iteratively increase resource utilization in a greedy fashion and to invoke a polynomial covering oracle that checks feasibility of every computed resource utilization. The approximation guarantee is best possible up to a constant factor. As a special case, our result 2) where the complexity of computing a socially optimal solution for matroid congestion games with non-decreasing cost functions is considered. Here, the approximation guarantee is best possible up to a constant factor if the number of resources is polynomially bounded in the number of players.

In Congestion Games, Taxes Achieve Optimal Approximation

“…Approximation algorithms. A number of polynomial time algorithms have been proposed for approximating the minimum social cost in congestion games and their network counterpart as discussed in [2,30,41] and references therein. The best known approximation is due to Makarychev and Sviridenko [41] who use randomization to round the solution of a natural linear programming relaxation.…”

Search for a moving target in a competitive environment

“…The interface between Operations Research and Game Theory is attracting much attention nowadays. Examples are (PPAD) computational complexity of Nash equilibria (see [25], [26], [22], [7], [29] and [8]), price of anarchy (see [28], [6]), congestion games (see [17], [16], [18] and [19]) and manufacturer-retailer problems (see [9] and [3]).…”

Uniqueness of Equilibria in Atomic Splittable Polymatroid Congestion Games

“…They also showed that the matroid property is the maximal property that gives rise to a pure Nash equilibrium, that is, for any strategy space not satisfying the matroid property, there is an instance of a weighted congestion game not having a pure Nash equilibrium. Integral polymatroid congestion games, a generalization of matroid congestion games, were later introduced in Harks, Klimm and Peis [15] (see also [16]). In addition, polymatroid theory was recently used in the context of nonatomic congestion games, where it is shown that matroid set systems are immune to the Braess paradox, see Fujishige et al [12].…”