Local search is a fundamental optimization technique that is both widely used in practice and deeply studied in theory, yet its computational complexity remains poorly understood. The traditional frameworks, PLS and the standard algorithm problem, introduced by Johnson, Papadimitriou, and Yannakakis (1988) fail to capture the methodology of local search algorithms: PLS is concerned with finding a local optimum and not with using local search, while the standard algorithm problem restricts each improvement step to follow a fixed pivoting rule. In this work, we introduce a novel formulation of local search which provides a middle ground between these models. In particular, the task is to output not only a local optimum but also a chain of local improvements leading to it. With this framework, we aim to capture the challenge in designing a good pivoting rule. Especially, when combined with the parameterized complexity paradigm, it enables both strong lower bounds and meaningful tractability results. Unlike previous works that combined parameterized complexity with local search, our framework targets the whole task of finding a local optimum and not only a single improvement step. Focusing on two representative meta-problems -- Subset Weight Optimization Problem with the $c$-swap neighborhood and Weighted Circuit with the flip neighborhood -- we establish fixed-parameter tractability results related to the number of distinct weights, while ruling out an analogous result when parameterized by the distance to the nearest optimum via a new type of reduction.

翻译：局部搜索作为一种基础优化技术，在实践中被广泛应用且在理论上得到深入研究，但其计算复杂性仍未得到充分理解。由Johnson、Papadimitriou和Yannakakis（1988）提出的传统框架——PLS与标准算法问题——未能准确刻画局部搜索算法的方法论：PLS关注于寻找局部最优解而非使用局部搜索过程，而标准算法问题则要求每个改进步骤必须遵循固定的枢轴规则。本研究提出了一种新颖的局部搜索形式化框架，在上述模型之间建立了折中方案。该框架要求不仅输出局部最优解，还需提供通向该解的局部改进链。通过此框架，我们旨在捕捉设计优良枢轴规则的核心挑战。特别地，当与参数化复杂性范式结合时，该框架既能支撑强下界证明，又能产生具有实际意义的可处理性结果。与以往将参数化复杂性与局部搜索相结合的研究不同，我们的框架针对寻找局部最优解的完整过程，而非仅关注单步改进。聚焦于两个代表性元问题——基于$c$-交换邻域的集合权重优化问题与基于翻转邻域的加权电路问题——我们建立了关于不同权重数量的固定参数可处理性结果，同时通过新型归约方法排除了基于距最近最优解距离参数化的类似可能性。