Online Makespan Scheduling under Scenarios

TL;DR

This work initiates a systematic study of online makespan scheduling under multiple known scenarios (OMSS), where a single online assignment must perform well across all scenario-restricted schedules. It extends Graham’s List Scheduling with new rules and a proxy-competitiveness framework, achieving a 5/3-competitive algorithm for m=2, K=2, and establishing near-tight lower bounds (≈1.64). For larger m and many scenarios, the paper shows a fundamental hardness: there exists a finite K_m after which no less-than-m-competitive deterministic online algorithm exists, via hypergraph-coloring-based constructions and hypertree gadgets. In the unit-processing-time setting, a 2-competitive algorithm for K=3 demonstrates a sharp contrast to the m-competitive lower bounds as K grows, and the results collectively map a rich landscape of how competitiveness deteriorates with more scenarios, with substantial implications for discrepancy minimization and online coloring.

Abstract

We consider a natural extension of online makespan scheduling on identical parallel machines by introducing scenarios. A scenario is a subset of jobs, and the task of our problem is to find a global assignment of the jobs to machines so that the maximum makespan under a scenario, i.e., the maximum makespan of any schedule restricted to a scenario, is minimized. For varying values of the number of scenarios and machines, we explore the competitiveness of online algorithms. We prove tight and near-tight bounds, several of which are achieved through novel constructions. In particular, we leverage the interplay between the unit processing time case of our problem and the hypergraph coloring problem both ways: We use hypergraph coloring techniques to steer an adversarial family of instances proving lower bounds, which in turn leads to lower bounds for several variants of online hypergraph coloring.

Figures (19)

• Figure 1: Deletion of the largest $j\in S_1\triangle S_2$ (checkered) does not increase the approximation ratio, as it does not affect the execution of Rule \ref{['asp:rulefix']}.
• Figure 2: Execution of the cutting lemma on job $n$, before (left) and after (right).
• Figure 3: Illustrative description of Observation \ref{['obs:bottleneck']}. Given the dark blue schedule, a worst outcome turns out to be the light blue job $n-1$ with a processing time $p_{n-1}$ such that $C_{n-1}=C_j$, followed by the light red job $n$ with $p_n=p(S_1\cap [n-1])$. The anticipation measures the minimum discrepancy in either machine prior to the revelation of the final job $n$.
• Figure 4: The first seven jobs in cases (i) and (ii) of the proof of Theorem \ref{['thm:53lb']}, respectively. In both cases, an eighth job $8\in S_1\cap S_2$ forces the schedules to the desired lower bound of competitiveness.
• Figure 5: The online schedule in the proof of Observation \ref{['obs:lbobs']} and the offline optimum.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 5
• Corollary 6
• Definition 9
• Definition 10
• Lemma 11: Deletion lemma, cf. Figure \ref{['fig:deletionlemma']}
• Lemma 12: Cutting lemma, cf. Figure \ref{['fig:cuttinglemma']}
• Definition 15