Non-Clairvoyant Scheduling is Hard Even for Trees

2026-10-07

AI Summary

This study addresses the challenge of non-clairvoyant online DAG parallel job scheduling, demonstrating that achieving a constant competitive ratio remains intractable even when jobs exhibit tree structures. Transcending the limitations of traditional FIFO policies, this work establishes a tight lower bound of Ω(min(m, OPT)) for any non-clairvoyant algorithm on tree-structured instances. Building upon this theoretical insight, it introduces a novel Small-Frontier-First strategy and designs a combinatorial greedy algorithm specifically optimized for scenarios with small OPT values. The primary contributions include delineating the fundamental theoretical limits of non-clairvoyant scheduling and presenting an algorithm that attains an asymptotically optimal competitive ratio. Collectively, these advances provide tight theoretical bounds and an efficient algorithmic framework for online DAG scheduling.

Abstract

We study online scheduling of parallel jobs on $m$ identical processors to minimize maximum flow time. Each arriving job is represented by a directed acyclic graph (DAG) whose vertices are unit-time subjobs and whose edges specify precedence constraints. We consider non-clairvoyant algorithms: the DAG is not known when a job arrives, and each subjob is revealed only when it becomes ready. Agrawal, Moseley, Newman, and Pruhs (SPAA 2024) showed that First-In-First-Out (FIFO) has competitive ratio $Ω(\log m)$ even when every job is an out-tree. They also proved that FIFO is $O(\log m)$-competitive in several natural settings and asked whether this guarantee extends to general instances. More broadly, they asked whether any non-clairvoyant algorithm can be $O(1)$-competitive. We answer both questions in the negative by proving a lower bound of $Ω(\min\{m,\mathrm{OPT}\})$ for every non-clairvoyant online algorithm, where $\mathrm{OPT}$ is the maximum flow time of an optimal offline schedule. In particular, every non-clairvoyant algorithm has competitive ratio $Ω(m)$ on some instance with $\mathrm{OPT} \ge m$. The lower bound holds even when every job is an out-forest. We complement these lower bounds with an asymptotically optimal non-clairvoyant algorithm. FIFO is $O(m)$-competitive, but can have competitive ratio $Ω(m)$ even when $\mathrm{OPT}=O(1)$. For instances with small $\mathrm{OPT}$, we design a Small-Frontier-First algorithm that is $O(\mathrm{OPT})$-competitive. Combining the two algorithms yields an $O(\min\{m,\mathrm{OPT}\})$-competitive non-clairvoyant algorithm.

Problem

• Non-Clairvoyant Scheduling
• Online Scheduling
• Maximum Flow Time
• Competitive Ratio
• Parallel Jobs

Innovation

• Non-Clairvoyant Scheduling
• Competitive Ratio
• Online Algorithms
• Maximum Flow Time
• Small-Frontier-First