We present a fully polynomial-time approximation scheme (FPTAS) for computing equilibria in congestion games, under \emph{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 \emph{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：我们提出了一个在全多项式时间近似方案（FPTAS）框架下计算拥塞博弈中均衡的算法，并采用平滑运行时间分析。具体而言，我们证明：若拥塞博弈的资源成本由独立噪声随机扰动，且噪声密度至多为$\phi$，则任意$(1+\varepsilon)$-改进动态序列将在期望步数内收敛至$(1+\varepsilon)$-近似纯纳什均衡（PNE），该步数关于$\frac{1}{\varepsilon}$、$\phi$及博弈描述规模呈强多项式增长。这一结果与传统最坏情形分析形成鲜明对比——在传统设定中，对于任意常数因子$\alpha\geq 1$，最优反应动态收敛至$\alpha$-近似PNE需要指数级时间。事实上，在拥塞博弈中计算$\alpha$-近似PNE属于PLS-难问题。我们展示了如何将分析推广至多种拥塞博弈模型，包括一般成本、阶梯函数成本、多项式成本以及公平成本分摊博弈（其中资源成本递减）。值得强调的是，我们的界值不显式依赖于玩家策略集的基数，因此该平滑FPTAS同样可直接应用于网络拥塞博弈。