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This paper concerns computing approximate pure Nash equilibria in weightedcongestion games, which has been shown to be PLS-complete. With the help of$\hat{\Psi}$-game and approximate potential functions, we propose twoalgorithms based on best response dynamics, and prove that they efficientlycompute $\frac{\rho}{1-\epsilon}$-approximate pure Nash equilibria for $\rho=d!$ and $\rho =\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1}\le {d + 1}$,respectively, when the weighted congestion game has polynomial latencyfunctions of degree at most $d \ge 1$ and players' weights are bounded fromabove by a constant $W \ge 1$. This improves the recent work of Feldotto etal.[2017] and Giannakopoulos et al. [2022] that showed efficient algorithms forcomputing $d^{d+o(d)}$-approximate pure Nash equilibria.

$\frac{\rho}{1-\epsilon}$-approximate pure Nash equilibria algorithms for weighted congestion games and their runtimes