Efficient computation of approximate pure Nash equilibria in congestion games

TL;DR

This paper introduces a simple polynomial-time algorithm for computing approximate pure Nash equilibria in congestion games, achieving near-optimal guarantees for linear and polynomial latency functions, and establishes PLS-completeness for certain deviations.

Contribution

It presents the first efficient algorithm for approximate equilibria in non-symmetric congestion games and proves PLS-completeness for broader cases.

Findings

Algorithm computes (2+ε)-approximate equilibria in polynomial time for linear latency functions.

Algorithm extends to polynomial latency functions with constant degree, with approximation guarantee d^{O(d)}.

Computing ρ-approximate equilibria is PLS-complete for certain 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 {\sf 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 $(2+\epsilon)$-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and $1/\epsilon$. 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 $\rho$-approximate equilibria is {\sf PLS}-complete for any polynomial-time computable $\rho$.