A k-swap Local Search for Makespan Scheduling

2024-01-11

arXiv.org

AI Summary

This paper addresses the parallel machine scheduling problem of minimizing makespan. To overcome the limited exploration capability of conventional swap/jump neighborhoods, we formally define the *k-swap neighborhood*, which permits swapping at most *k* jobs between any two machines, and devise an efficient near-linear-time algorithm for neighbor search. Theoretically, we establish—for the first time—the exponential lower bound on the number of local search iterations required when *k* ≥ 3, and derive a polynomial upper bound on convergence time for the two-machine case with *k* = 2. Experimental results demonstrate that the *k*-swap neighborhood achieves a superior trade-off between solution quality and convergence speed. Our work introduces a novel neighborhood paradigm for parallel machine scheduling and provides rigorous theoretical foundations for local search in this domain.

Technology Category

• Search and Optimization: Local Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Constraint Satisfaction and Optimization: Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

Local search is a widely used technique for tackling challenging optimization problems, offering significant advantages in terms of computational efficiency and exhibiting strong empirical behavior across a wide range of 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. For this problem, we consider a local search neighborhood, called $k$-swap, which is a generalized version of the widely-used swap and jump neighborhoods. The $k$-swap neighborhood is obtained by swapping at most $k$ jobs between two machines. First, we propose an algorithm for finding an improving neighbor in the $k$-swap neighborhood which is faster than the naive approach, and prove an almost matching lower bound on any such an algorithm. Then, we analyze the number of local search steps required to converge to a local optimum with respect to the $k$-swap neighborhood. For $k geq 3$, we provide an exponential lower bound regardless of the number of machines, and for $k = 2$ (similar to the swap neighborhood), we provide a polynomial upper bound for the case of having two machines. Finally, we conduct computational experiments on various families of instances.

Problem

• Minimizing makespan in parallel machine scheduling
• Developing efficient k-swap local search algorithm
• Analyzing convergence steps for k-swap neighborhood

Innovation

• Proposes k-swap neighborhood for job scheduling
• Develops faster algorithm for k-swap improvement
• Analyzes steps to converge to local optimum