Efficient Computation of Approximate Pure Nash Equilibria in Congestion Games

Abstract

Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general PLS-complete. We present a surprisingly simple polynomial-time algorithm that computes O(1)-approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes (2 + ǫ)-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and 1/ǫ. It also applies to games with polynomial latency functions with constant maximum degree d; there, the approximation guarantee is d O(d) . The algorithm essentially identifies a polynomially long sequence of best-response moves that lead to an approximate equilibrium; the existence of such short sequences is interesting in itself. These are the first positive algorithmic results for approximate equilibria in non-symmetric congestion games. We strengthen them further by proving that, for congestion games that deviate from our mild assumptions, computing ρ-approximate equilibria is PLS-complete for any polynomial-time computable ρ.

Computing approximate pure Nash equilibria in congestion games

“…Our algorithm for unweighted congestion games is presented in [Caragiannis et al 2011]. It computes (2 + )-approximate pure Nash equilibria in games with linear latency functions, and d d+o(d) approximate equilibria for polynomial latency functions of maximum degree d. The algorithm is surprisingly simple.…”

“…In two recent papers [Caragiannis et al 2011;2012], we present algorithms for computing O(1)-approximate equilibria in unweighted and weighted non-symmetric congestion games with polynomial latency functions of constant maximum degree. Our algorithm for unweighted congestion games is presented in [Caragiannis et al 2011].…”

“…For example, the stretch is almost 2 for linear unweighted congestion games, 3+ √ 5 2 for linear weighted congestion games, and d d+o(d) for Ψ-games of degree d ≥ 2. A more thorough literature review, the detailed description of the algorithms, and the analysis details can be found in [Caragiannis et al 2011;2012].…”

On the Complexity of Nash Equilibria in Anonymous Games

“…[PR08] gave a polynomial-time for finding an exact Nash equilibrium in a symmetric game. For congestion games, PLS-completeness of pure equilibria was established in [FPT04,ARV08,SV08] 3 , and efficient approximation algorithms for various latency functions were obtained in [CFGS11,CFGS12,CS11].…”

Totally Unimodular Congestion Games

“…In this work, we focus on congestion games, a class of potential games that has been widely investigated in the literature (see, e.g., [18,20,26,7,19,4,5]), and we provide new insight on their computational complexity using a polyhedral approach. In a congestion game, a set of resources is given, and each player selects a feasible subset of the resources in order to minimize her cost function.…”