Smoothed analysis of the k-swap neighborhood for makespan scheduling

Abstract

In this paper, we address the problem of scheduling a set of n jobs on m identical parallel machines with the objective of makespan minimization, by considering a local search neighborhood, called k-swap. In our previous study, we provided an exponential lower bound of $2^{\Omega \left(n\right)}$ for $k\ge 3$. In this study, we show that the smoothed number of iterations in finding a local optimum with respect to the k-swap neighborhood is $O(m^2\cdot n^{2k+2}\cdot \log m\cdot \varphi )$, where $\varphi \ge 1$ is the perturbation parameter.