Smoothed Analysis of the k-Swap Neighborhood for Makespan Scheduling

TL;DR

This paper applies smoothed analysis to the $k$-swap local search for makespan scheduling, showing that the expected number of iterations to reach a local optimum is polynomially bounded, explaining its practical efficiency.

Contribution

The paper provides the first smoothed analysis bound for the $k$-swap local search in makespan scheduling, bridging the gap between worst-case and practical performance.

Findings

Smoothed number of iterations is bounded by $O(m^2 ^{2k+2} \, \log m \, \phi)$.

The analysis explains why $k$-swap local search performs well in practice despite poor worst-case bounds.

The bound indicates that worst-case scenarios are rare under smoothed analysis.

Abstract

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.