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：我们提出了一种在光滑运行时间分析下，用于计算拥塞博弈中平衡的全多项式时间近似方案（FPTAS）。更精确地说，我们证明：如果拥塞博弈的资源成本被密度至多为$\phi$的独立噪声随机扰动，那么任何$(1+\varepsilon)$-改进动态序列都将期望在强多项式于$\frac{1}{\varepsilon}$、$\phi$及博弈描述规模步数内达到一个$(1+\varepsilon)$-近似纯纳什均衡（PNE）。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，已知对于任意常数因子$\alpha\geq 1$，最优反应动态收敛到$\alpha$-近似PNE所需时间呈指数级增长。事实上，在拥塞博弈中计算$\alpha$-近似PNE是PLS-困难的。我们演示了如何将分析应用于多种不同拥塞博弈模型，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）。值得注意的是，我们的界不显式依赖于玩家策略集的基数，因此光滑FPTAS也可直接应用于网络拥塞博弈。