We present a deterministic polynomial-time algorithm for computing $d^{d+o(d)}$-approximate (pure) Nash equilibria in weighted congestion games with polynomial cost functions of degree at most $d$. This is an exponential improvement of the approximation factor with respect to the previously best algorithm. An appealing additional feature of our algorithm is that it uses only best-improvement steps in the actual game, as opposed to earlier approaches that first had to transform the game itself. Our algorithm is an adaptation of the seminal algorithm by Caragiannis et al. [FOCS'11, TEAC 2015], but we utilize an approximate potential function directly on the original game instead of an exact one on a modified game. A critical component of our analysis, which is of independent interest, is the derivation of a novel bound of $[d/\mathcal{W}(d/\rho)]^{d+1}$ for the Price of Anarchy (PoA) of $\rho$-approximate equilibria in weighted congestion games, where $\mathcal{W}$ is the Lambert-W function. More specifically, we show that this PoA is exactly equal to $\Phi_{d,\rho}^{d+1}$, where $\Phi_{d,\rho}$ is the unique positive solution of the equation $\rho (x+1)^d=x^{d+1}$. Our upper bound is derived via a smoothness-like argument, and thus holds even for mixed Nash and correlated equilibria, while our lower bound is simple enough to apply even to singleton congestion games.

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