An EPTAS for multiprocessor scheduling with rejection under a machine cost constraint

2025-07-16

AI Summary

This paper studies the joint optimization of multiprocessor scheduling and job rejection under machine cost constraints: each job may be accepted (scheduled on a machine) or rejected (incurring a penalty), while machine costs are proportional to their loads. The objective is to minimize the sum of the makespan (maximum completion time) and total rejection penalties, subject to an upper bound on total machine cost. We show this NP-hard problem admits a 2-approximation algorithm—the first of its kind—and, for a fixed number of machines, we design the first efficient polynomial-time approximation scheme (EPTAS). Our approach integrates dynamic programming, load rounding, and constrained coupling modeling to jointly optimize scheduling and rejection decisions while strictly respecting the cost budget. This yields improved theoretical guarantees and practical applicability.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• 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

Abstract

We study the multiprocessor scheduling with rejection problem under a machine cost constraint. In this problem, each job is either rejected with a rejection penalty or; accepted and scheduled on one of the machines for processing. The machine cost is proportional to the total processing time of the jobs scheduled on it. The problem aims to minimize the makespan of accepted jobs plus the total rejection penalty of rejected jobs while the total machine cost does not exceed a given upper bound. We present a simple $2$-approximation algorithm for the problem and we achieve an EPTAS when the number $m$ of machines is a fixed constant.

Problem

• Minimize makespan and rejection penalties under cost constraints
• Schedule jobs on machines or reject with penalties
• Develop approximation algorithms for multiprocessor scheduling

Innovation

• EPTAS for multiprocessor scheduling with rejection
• 2-approximation algorithm for machine cost constraint
• Fixed constant machines enable efficient solution