We propose a unifying framework for smoothed analysis of combinatorial local optimization problems, and show how a diverse selection of problems within the complexity class PLS can be cast within this model. This abstraction allows us to identify key structural properties, and corresponding parameters, that determine the smoothed running time of local search dynamics. We formalize this via a black-box tool that provides concrete bounds on the expected maximum number of steps needed until local search reaches an exact local optimum. This bound is particularly strong, in the sense that it holds for any starting feasible solution, any choice of pivoting rule, and does not rely on the choice of specific noise distributions that are applied on the input, but it is parameterized by just a global upper bound $\phi$ on the probability density. The power of this tool can be demonstrated by instantiating it for various PLS-hard problems of interest to derive efficient smoothed running times (as a function of $\phi$ and the input size). Most notably, we focus on the important local optimization problem of finding pure Nash equilibria in Congestion Games, that has not been studied before from a smoothed analysis perspective. Specifically, we propose novel smoothed analysis models for general and Network Congestion Games, under various representations, including explicit, step-function, and polynomial resource latencies. We study PLS-hard instances of these problems and show that their standard local search algorithms run in polynomial smoothed time. Finally, we present further applications of our framework to a wide range of additional combinatorial problems, including local Max-Cut in weighted graphs, the Travelling Salesman problem (TSP) under the $k$-opt local heuristic, and finding pure equilibria in Network Coordination Games.

翻译：本文提出了一个用于组合局部优化问题平滑分析的统一框架，并展示了如何将复杂度类PLS中的多种问题纳入该模型。该抽象框架使我们能够识别决定局部搜索动态平滑运行时间的关键结构特性及相关参数。我们通过一个黑盒工具对此进行形式化，该工具可为局部搜索达到精确局部最优解所需的最大期望步数提供具体边界。该边界具有特别强的普适性：它适用于任意初始可行解、任意枢轴规则选择，且不依赖于输入所施加的特定噪声分布选择，仅通过概率密度的全局上界$\phi$进行参数化。通过将该工具实例化应用于多个重要的PLS难问题，可推导出高效的平滑运行时间（作为$\phi$和输入规模的函数），从而展示此工具的威力。我们特别关注寻找拥堵博弈中纯纳什均衡这一重要局部优化问题——该问题此前尚未从平滑分析视角进行研究。具体而言，我们针对广义及网络拥堵博弈提出了新颖的平滑分析模型，涵盖显式表示、阶梯函数及多项式资源延迟等多种表示形式。通过研究这些问题的PLS难实例，我们证明其标准局部搜索算法在多项式平滑时间内运行。最后，我们进一步展示了该框架在多种组合问题中的广泛应用，包括加权图中的局部最大割问题、基于$k$-opt局部启发式的旅行商问题（TSP），以及网络协调博弈中纯均衡的求解。