We present a fully polynomial-time approximation scheme (FPTAS) for computing equilibria in congestion games, under smoothed running-time analysis. More precisely, we prove that if the resource costs of a congestion game are randomly perturbed by independent noises, whose density is at most $\phi$, then any sequence of $(1+\varepsilon)$-improving dynamics will reach an $(1+\varepsilon)$-approximate pure Nash equilibrium (PNE) after an expected number of steps which is strongly polynomial in $\frac{1}{\varepsilon}$, $\phi$, and the size of the game's description. Our results establish a sharp contrast to the traditional worst-case analysis setting, where it is known that better-response dynamics take exponentially long to converge to $\alpha$-approximate PNE, for any constant factor $\alpha\geq 1$. As a matter of fact, computing $\alpha$-approximate PNE in congestion games is PLS-hard. We demonstrate how our analysis can be applied to various different models of congestion games including general, step-function, and polynomial cost, as well as fair cost-sharing games (where the resource costs are decreasing). It is important to note that our bounds do not depend explicitly on the cardinality of the players' strategy sets, and thus the smoothed FPTAS is readily applicable to network congestion games as well.

翻译：我们提出了一个用于计算拥堵博弈中均衡的全多项式时间近似方案（FPTAS），并采用了光滑运行时间分析。更精确地说，我们证明如果拥堵博弈的资源成本受到独立噪声的随机扰动，且噪声密度至多为 $\phi$，那么任何 $(1+\varepsilon)$-改进动态序列在期望步数内将达到一个 $(1+\varepsilon)$-近似纯纳什均衡（PNE），该步数在 $\frac{1}{\varepsilon}$、$\phi$ 以及博弈描述规模上是强多项式的。我们的结果与传统最坏情况分析设定形成鲜明对比，在传统设定中，已知对于任何常数因子 $\alpha\geq 1$，最佳响应动态收敛到 $\alpha$-近似PNE需要指数级长时间。事实上，计算拥堵博弈中的 $\alpha$-近似PNE是PLS-hard的。我们展示了如何将我们的分析应用于各种不同的拥堵博弈模型，包括一般成本、阶梯函数成本和多项式成本，以及公平成本分摊博弈（其中资源成本递减）。值得注意的是，我们的界限不显式依赖于玩家策略集合的基数，因此光滑FPTAS同样适用于网络拥堵博弈。