Efficient computation of approximate pure Nash equilibria in congestion games

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 $\rho$.

💡 Research Summary

The paper addresses the computational difficulty of finding pure Nash equilibria in congestion games, a problem known to be PLS‑complete in general. Recognizing that exact equilibria are often infeasible to compute, the authors focus on approximate equilibria, where no player can improve her cost by more than a factor ρ. They present a surprisingly simple polynomial‑time algorithm that, for a broad class of congestion games with non‑negative polynomial latency functions, computes an O(1)‑approximate pure Nash equilibrium.

The algorithm starts from a specific initial state in which every player plays her best response assuming no other players are present. It then proceeds in a series of “phases”. In phase k a cost threshold interval