Almost Tight Approximation Hardness and Online Algorithms for Resource Scheduling

2025-08-31

AI Summary

This paper investigates job scheduling with resource constraints and precedence dependencies, aiming to minimize the makespan subject to a global resource budget. For both offline and online settings, it establishes the fundamental computational complexity of the problem—first revealing a deep connection to the Shortest Common Supersequence problem—and proves an $o((log t_{max})^alpha)$ inapproximability bound for the offline case. For the online setting, it derives tight lower bounds of $Omega(log n)$ and $Omega(log t_{max})$ on the competitive ratio. Furthermore, it proposes a deterministic online algorithm matching these lower bounds, thereby achieving tight theoretical guarantees. Collectively, these results provide a unified characterization of the approximability limits and tractability boundaries for resource-aware scheduling.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the precedence-constrained resource scheduling problem [SICOMP'75]. There are $n$ jobs where each job takes a certain time to finish and has a resource requirement throughout the execution time. There are precedence among the jobs. The problem asks that given a resource budget, schedule the jobs obeying the precedence constraints to minimize makespan (maximum completion time of a job) such that at any point in time, the total resource being used by all the jobs is at most the given resource budget. In the offline setting, an important open question is whether a polynomial-time $O(1)$-factor approximation algorithm can be found. We prove almost tight hardness of approximation: For some constant $α> 0$, there is no $o((log t_{max})^α)$-factor ( or $o( ( log n )^α)$-factor ) approximation algorithm with $n$ jobs of maximum job length $t_{max}$, unless P = NP ( or NP $subset$ DTIME$(O( 2^{ ext{polylog}(n)}))$ ). We further show a connection between this scheduling problem and a seemingly unrelated problem called the shortest common super-sequence (SCS) problem, which has wide application in Biology and Genomics. We prove that an $o(log t_{max})$-factor approximation of the scheduling problem would imply the existence of an $o(|Σ|)$-approximation algorithm for SCS with alphabet $Σ$. We then consider the online setting. We present $Ω(log n)$ and $Ω(log t_{max})$ lower bounds of the competitive ratio of any randomized online algorithm. Moreover, we present a matching $O(min{log n, log t_{max}})$-competitive deterministic online algorithm.

Problem

• Studying precedence-constrained resource scheduling with makespan minimization
• Proving almost tight hardness of approximation for offline algorithms
• Establishing competitive ratio bounds for online scheduling algorithms

Innovation

• Proved almost tight approximation hardness bounds
• Connected scheduling to shortest common super-sequence problem
• Developed matching deterministic online algorithm