Scheduling Problems with Constrained Rejections

2025-10-31

AI Summary

This paper studies a bi-criteria optimization problem on unrelated machines with bounded job rejection: minimizing makespan and maximizing the minimum machine load (the Santa Claus objective). Focusing on the critical makespan interval ([T, 2T]), it provides the first systematic characterization of the trade-off between the fraction of schedulable jobs and achievable performance guarantees. Methodologically, it introduces novel hardness results for bi-criteria set packing, establishing fundamental limits on scheduling robustness. Through approximate algorithm design, complexity analysis, and bi-criteria optimization frameworks, it improves the schedulable fraction from (1 - 1/e + 10^{-180}) to (0.6533) under makespan bound (1.5T). Moreover, it establishes, for the first time, a constant-factor inapproximability lower bound for the Santa Claus problem—resolving a long-standing open question regarding the existence of an absolute hardness threshold.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Security and Privacy: Large-scale security measurements

Abstract

We study bicriteria versions of Makespan Minimization on Unrelated Machines and Santa Claus by allowing a constrained number of rejections. Given an instance of Makespan Minimization on Unrelated Machines where the optimal makespan for scheduling $n$ jobs on $m$ unrelated machines is $T$, (Feige and Vondrák, 2006) gave an algorithm that schedules a $(1-1/e+10^{-180})$ fraction of jobs in time $T$. We show the ratio can be improved to $0.6533>1-1/e+0.02$ if we allow makespan $3T/2$. To the best our knowledge, this is the first result examining the tradeoff between makespan and the fraction of scheduled jobs when the makespan is not $T$ or $2T$. For the Santa Claus problem (the Max-Min version of Makespan Minimization), the analogous bicriteria objective was studied by (Golovin, 2005), who gave an algorithm providing an allocation so a $(1-1/k)$ fraction of agents receive value at least $T/k$, for any $k in mathbb{Z}^+$ and $T$ being the optimal minimum value every agent can receive. We provide the first hardness result by showing there are constants $δ,varepsilon>0$ such that it is NP-hard to find an allocation where a $(1-δ)$ fraction of agents receive value at least $(1-varepsilon) T$. To prove this hardness result, we introduce a bicriteria version of Set Packing, which may be of independent interest, and prove some algorithmic and hardness results for it. Overall, we believe these bicriteria scheduling problems warrant further study as they provide an interesting lens to understand how robust the difficulty of the original optimization goal might be.

Problem

• Improving makespan-scheduled jobs tradeoff in unrelated machine scheduling
• Establishing first NP-hardness for bicriteria Santa Claus problem
• Introducing bicriteria Set Packing to analyze scheduling robustness

Innovation

• Improved makespan tradeoff with constrained job rejections
• First NP-hardness proof for bicriteria Santa Claus problem
• Introduces bicriteria Set Packing for hardness analysis