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. We then demonstrate the power of this tool by instantiating it for various PLS-hard problems to derive efficient smoothed running times. This not only unifies, and greatly simplifies, prior existing positive results, but also allows us to extend or improve them. Notable problems on which we provide such a contribution are Max-Cut, the Travelling Salesman problem, and Network Coordination Games. Additionally, in this paper we propose novel smoothed analysis formulations, and prove polynomial smoothed running times, for important local optimization problems that have not been studied before from this perspective. Importantly, we provide an extensive study of the problem of finding pure Nash equilibria in general and Network Congestion Games under various representation models, including explicit, step-function, and polynomial latencies. We show that all the problems we study can be solved by their standard local search algorithms in polynomial smoothed time on PLS-hard instances in which these algorithms have exponential worst-case running time.

翻译：我们提出一个用于组合局部优化问题平滑分析的统一框架，并展示复杂度类PLS中多种不同问题如何纳入此模型。该抽象方法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性质及其对应参数。我们通过一个黑箱工具形式化这一方法，该工具可为局部搜索达到精确局部最优所需的期望最大步数提供具体界限。该界限具有特别强的适用性——既适用于任意初始可行解与任意枢轴规则，也不依赖于输入噪声分布的具体选择，而仅由概率密度的全局上界φ参数化。我们通过将该工具应用于各类PLS-难问题来证明其效力，从而推导出高效的平滑运行时间。这不仅统一并大幅简化了已有正面结论，还使我们能够扩展或改进这些结果。我们提供此类贡献的著名问题包括最大割、旅行商问题和网络协调博弈。此外，本文针对此前未从平滑分析角度研究的重要局部优化问题，提出了新型平滑分析框架并证明了多项式平滑运行时间。重要的是，我们对一般纳什均衡和网络拥塞博弈在显式、阶梯函数及多项式延迟等多种表示模型下的纯策略纳什均衡求解问题进行了全面研究。结果表明，我们研究的所有问题均可通过标准局部搜索算法在多项式平滑时间内解决，即使这些算法在PLS-难实例上具有指数级最坏情况运行时间。