An Efficient PTAS for Parallel Machine Scheduling with Capacity Constraints

• Published in: Combinatorial Optimization and Applications Vol. 10043; pp. 608 - 623
• Main Authors: Chen, Lin, Jansen, Klaus, Luo, Wenchang, Zhang, Guochuan
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2016; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Algorithms & data structures; Approximation algorithms; Capacity constraints; Discrete mathematics; Integer programming; Mathematical theory of computation; Scheduling
• ISBN: 9783319487489; 3319487485
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-48749-6_44

In this paper, we consider the classical scheduling problem on parallel machines with capacity constraints. We are given m identical machines, where each machine k can process up to $$c_k$$ jobs. The goal is to assign the $$n\le \sum _{k=1}^{m}c_k$$ independent jobs on the machines subject to the capacity constraints such that the makespan is minimized. This problem is a generalization of the c-partition problem where $$c_k=c$$ for each machine. The c-partition problem is strongly NP-hard for $$c\ge 3$$ and the best known approximation algorithm of which has a performance ratio of 4 / 3 due to Babel et al. [2]. For the general problem where machines could have different capacities, the best known result is a 1.5-approximation algorithm with running time $$O(n\log n+m^2n)$$ [14]. In this paper, we improve the previous result substantially by establishing an efficient polynomial time approximation scheme (EPTAS). The key idea is to establish a non-standard ILP (Integer Linear Programming) formulation for the scheduling problem, where a set of crucial constraints (called proportional constraints) is introduced. Such constraints, along with a greedy rounding technique, allow us to derive an integer solution from a relaxed fractional one without violating constraints.