Delayed-Clairvoyant Flow Time Scheduling via a Borrow Graph Analysis

TL;DR

This paper proposes a scheduling rule with a competitive ratio of $\mathcal{O}(\frac{1}{1-\alpha})$ whenever $0 \leq \alpha<1$.

Abstract

We study the problem of preemptively scheduling jobs online over time on a single machine to minimize the total flow time. In the traditional clairvoyant scheduling model, the scheduler learns about the processing time of a job at its arrival, and scheduling at any time the job with the shortest remaining processing time (SRPT) is optimal. In contrast, the practically relevant non-clairvoyant model assumes that the processing time of a job is unknown at its arrival, and is only revealed when it completes. Non-clairvoyant flow time minimization does not admit algorithms with a constant competitive ratio. Consequently, the problem has been studied under speed augmentation (JACM'00) or with predicted processing times (STOC'21, SODA'22) to attain constant guarantees. In this paper, we consider $α$-clairvoyant scheduling, where the scheduler learns the processing time of a job once it completes an $α$-fraction of its processing time. This naturally interpolates between clairvoyant scheduling ($α=0$) and non-clairvoyant scheduling ($α=1$). By elegantly fusing two traditional algorithms, we propose a scheduling rule with a competitive ratio of $\mathcal{O}(\frac{1}{1-α})$ whenever $0 \leq α< 1$. As $α$ increases, our competitive guarantee transitions nicely (up to constants) between the previously established bounds for clairvoyant and non-clairvoyant flow time minimization. We complement this positive result with a tight randomized lower bound.

Figures (8)

• Figure 1: Situation of our algorithm where the simple matching argument does not work.
• Figure 2: Example of a job $j \in D(t)$ and $k \in N(t)$ with a bad edge $(j,k) \in E_N$. At point in time $s_j$, the jobs satisfy $y_j(s_j) = y_k(s_j)$ as they are processed in parallel. However, as $k$ is executed during $[s_j,t]$, we have $\bm\bar{y}_j = y_j(s_j) = y_k(s_j) < y_k(t) = \bm\bar{y}_k$, and the edge is bad.
• Figure 3: Illustrates the possible values of $t'$ in the proof of \ref{['lem:layers:entrypoints:1']}: The red area indicates values that are not possible while the green area indicates values that are indeed possible.
• Figure 4: The situation considered in \ref{['lem:multiple:entries:1']}.
• Figure 5: Illustrates the possible (green) and impossible (red) values of $s_{k_2}$ in the proof of \ref{['lem:multiple:entries:1']}

Theorems & Definitions (153)

• Theorem 1.1
• Theorem 1.2
• proof
• Theorem 3.1
• Definition 3.2: Lifetime
• Definition 3.3: Borrowing
• Definition 3.4: Borrow Graph
• Lemma 3.5
• proof
• Corollary 3.6