组合局部搜索的平滑复杂度研究
On the Smoothed Complexity of Combinatorial Local Search
AI总结：
本文提出组合局部优化问题平滑分析的统一框架，通过黑盒工具给出仅依赖概率密度上界$ϕ$的局部搜索步数上界，并首次对拥塞博弈的纯纳什均衡求解进行平滑分析，证明其标准局部搜索算法在多项式平滑时间内运行。
摘要：
我们提出了一个用于组合局部优化问题平滑分析的统一框架，并展示了复杂度类PLS中的多种问题如何被纳入该模型。这一抽象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相应参数。我们通过一个黑盒工具对此进行形式化，该工具为局部搜索达到精确局部最优所需的最大期望步数提供了具体上界。该上界特别强，因为它对任意起始可行解、任意枢轴规则选择均成立，并且不依赖于施加在输入上的特定噪声分布选择，而仅由一个关于概率密度的全局上界$ϕ$参数化。该工具的能力可以通过将其实例化到各种PLS难问题上来展示，从而推导出高效的平滑运行时间（作为$ϕ$和输入规模的函数）。最值得注意的是，我们聚焦于在拥塞博弈中寻找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的角度被研究过。具体而言，我们针对一般拥塞博弈和网络拥塞博弈，在包括显式、阶跃函数和多项式资源延迟在内的多种表示下，提出了新的平滑分析模型。我们研究了这些问题的PLS难实例，并证明其标准局部搜索算法在多项式平滑时间内运行。最后，我们展示了该框架在更广泛的组合问题上的进一步应用，包括加权图中的局部最大割问题、$k$-opt局部启发式下的旅行商问题（TSP），以及网络协调博弈中纯均衡的求解。
英文摘要：
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 $ϕ$ 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 $ϕ$ 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.
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