Local search is a widely used technique for tackling challenging optimization problems, offering simplicity and strong empirical performance across various problem domains. In this paper, we address the problem of scheduling a set of jobs on identical parallel machines with the objective of makespan minimization, by considering a local search neighborhood, called $k$-swap. A $k$-swap neighbor is obtained by interchanging the machine allocations of at most $k$ jobs scheduled on two machines. While local search algorithms often perform well in practice, they can exhibit poor worst-case performance. In our previous study, we showed that for $k \geq 3$, there exists an instance where the number of iterations required to converge to a local optimum is exponential in the number of jobs. Motivated by this discrepancy between theoretical worst-case bound and practical performance, we apply smoothed analysis to the $k$-swap local search. Smoothed analysis has emerged as a powerful framework for analyzing the behavior of algorithms, aiming to bridge the gap between poor worst-case and good empirical performance. In this paper, we show that the smoothed number of iterations required to find a local optimum with respect to the $k$-swap neighborhood is bounded by $O(m^2 \cdot n^{2k+2} \cdot \log m \cdot \phi)$, where $n$ and $m$ are the numbers of jobs and machines, respectively, and $\phi \geq 1$ is the perturbation parameter. The bound on the smoothed number of iterations demonstrates that the proposed lower bound reflects a pessimistic scenario which is rare in practice.

翻译：局部搜索是解决复杂优化问题的一种广泛应用的技术，以其简单性和在不同问题领域中表现出的强大实证性能而著称。本文通过考虑一种称为$k$-交换的局部搜索邻域，研究了在相同并行机器上调度一组作业以最小化最大完工时间的问题。$k$-交换邻域解是通过交换两台机器上最多$k$个作业的机器分配而得到的。虽然局部搜索算法在实践中通常表现良好，但其最坏情况性能可能较差。在我们先前的研究中，我们证明了对于$k \geq 3$，存在一个实例，其收敛到局部最优解所需的迭代次数关于作业数量是指数级的。受理论最坏情况界与实证性能之间差异的启发，我们将平滑分析应用于$k$-交换局部搜索。平滑分析已成为分析算法行为的一个强大框架，旨在弥合较差的最坏情况性能与良好的实证性能之间的差距。本文中，我们证明了在$k$-交换邻域下找到局部最优解所需的平滑迭代次数以$O(m^2 \cdot n^{2k+2} \cdot \log m \cdot \phi)$为界，其中$n$和$m$分别是作业和机器的数量，$\phi \geq 1$是扰动参数。平滑迭代次数的界表明，所提出的下界反映的是一种在实践中罕见的悲观情形。