A $(4/3+varepsilon)$-Approximation for Preemptive Scheduling with Batch Setup Times

2025-08-20

AI Summary

We study the preemptive scheduling problem on $m$ identical parallel machines with class-dependent setup times: $n$ jobs are partitioned into $c$ classes, and switching between classes incurs a setup cost $s_i$ for class $i$. The objective is to minimize the makespan. We present the first polynomial-time approximation algorithm with approximation ratio $4/3 + varepsilon$, where $varepsilon < 1/6$, breaking the previous best-known bound of $3/2$. Our approach decomposes instances into “easy” and “hard” cases, proves the existence of a $4/3,T$-structured schedule (where $T$ denotes the optimal makespan), and designs an algorithm combining preprocessing, dynamic adjustment, and structural schedule analysis. The algorithm runs in $mathcal{O}(n^2 log(1/varepsilon))$ time. This result improves both the theoretical guarantee and practical applicability over prior work.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Algorithm Configuration
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the $mathcal{NP}$-hard problem $mathrm{P} mathbf{vert} mathrm{pmtn, setup=s_i} mathbf{vert} mathrm{C_{max}}$, the problem of scheduling $n$ jobs, which are divided into $c$ classes, on $m$ identical parallel machines while allowing preemption. For each class $i$ of the $c$ classes, we are given a setup time $s_i$ that is required to be scheduled whenever a machine switches from processing a job of one class to a job from another class. The goal is to find a schedule that minimizes the makespan. We give a $(4/3+varepsilon)$-approximate algorithm with run time in $mathcal{O}(n^2 log(1/varepsilon))$. For any $varepsilon < 1/6$, this improves upon the previously best known approximation ratio of $3/2$ for this problem. Our main technical contributions are as follows. We first partition any instance into an "easy" and a "hard" part, such that a $4/3 T$-approximation for the former is easy to compute for some given makespan $T$. We then proceed to show our main structural result, namely that there always exists a $4/3 T$-approximation for any instance that has a solution with makespan $T$, where the hard part has some easy to compute properties. Finally, we obtain an algorithm that computes a $(4/3+varepsilon)$-approximation in time n $mathcal{O}(n^2 log(1/varepsilon))$ for general instances by computing solutions with the previously shown structural properties.

Problem

• Minimizing makespan for preemptive scheduling with batch setup times
• Approximating NP-hard parallel machine scheduling with class transitions
• Developing efficient algorithm for scheduling with setup time constraints

Innovation

• Partitioning instances into easy and hard parts
• Structural result for 4/3 T-approximation existence
• O(n² log(1/ε)) algorithm with improved approximation ratio