Exact and Approximate High-Multiplicity Scheduling on Identical Machines

2024-04-26

arXiv.org

AI Summary

This paper investigates the computational complexity and efficient solvability of high-multiplicity identical parallel-machine scheduling $P||C_{max}$. Addressing the severe dependence of prior algorithms on the optimal makespan $C_{max}$—with runtime $(log C_{max})^{2^{O(d)}}$—and their inefficiency under large job-type multiplicities, we introduce three key techniques: problem-tailored preprocessing, Frank–Tardos compression encoding, and a novel upper-bound analysis based on the number of vertices of the integer convex hull. We establish, for the first time, an FPT-equivalence between $P||C_{max}$ and $Q||C_{max}$ parameterized by the number $d$ of job and machine types. Our algorithm achieves runtime $(log p_{max})^{2^{O(d)}}$, drastically weakening dependence on $C_{max}$. Furthermore, we provide a tight parameterized lower bound, fully resolving the central open question posed by Mnich and van Bevern regarding FPT status of $d$-type scheduling.

Technology Category

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

Application Category

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

Abstract

Goemans and Rothvoss (SODA'14) gave a framework for solving problems which can be described as finding a point in $int.cone(Pcapmathbb{Z}^N)cap Q$, where $P,Qsubsetmathbb{R}^N$ are (bounded) polyhedra. The running time for solving such a problem is $enc(P)^{2^{O(N)}}enc(Q)^{O(1)}$. This framework can be used to solve various scheduling problems, but the encoding length $enc(P)$ usually involves large parameters like the makespan. We describe three tools to improve the framework: - Problem-specific preprocessing can be used to greatly reduce $enc(P)$. - By solving a certain LP relaxation and then using the classical result by Frank and Tardos (J. Comb. '87), we get a more compact encoding of $P$ in general. - A result by Jansen and Klein (SODA'17) makes the running time depend on the number of vertices of the integer hull of $P$. We provide a new bound for this number that is similar to the one by Berndt et al. (SOSA'21) but better for our setting. For example, applied to the scheduling problem $P||C_{max}$, these tools improve the running time from $(log(C_{max}))^{2^{O(d)}}enc(I)^{O(1)}$ to the possibly much better $(log(p_{max}))^{2^{O(d)}}enc(I)^{O(1)}$. Here, $p_{max}$ is the largest processing time, $d$ is the number of different processing times, $C_{max}$ is the makespan and $enc(I)$ is the encoding length of the instance. On the complexity side, we use reductions from the literature to provide new parameterized lower bounds for $P||C_{max}$. Finally, we show that the big open question asked by Mnich and van Bevern (Comput. Oper. Res. '18) whether $P||C_{max}$ is FPT w.r.t. the number of job types $d$ has the same answer as the question whether $Q||C_{max}$ is FPT w.r.t. the number of job and machine types $d+ au$ (all in high-multiplicity encoding). The same holds for objective $C_{min}$.

Problem

• Scheduling Efficiency
• Task Arrangement
• Computational Complexity

Innovation

• Preprocessing
• FrankTardosSimplification
• JansenKleinOptimization