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A Simpler Analysis for $\varepsilon$-Clairvoyant Flow Time Scheduling

Anupam Gupta, Haim Kaplan, Alexander Lindermayr, Jens Schlöter, Sorrachai Yingchareonthawornchai

Abstract

We simplify the proof of the optimality of the Shortest Lower-Bound First (SLF) algorithm, introduced by Gupta, Kaplan, Lindermayr, Schlöter, and Yingchareonthawornchai [FOCS'25], for minimizing the total flow time in the $\varepsilon$-clairvoyant setting.

A Simpler Analysis for $\varepsilon$-Clairvoyant Flow Time Scheduling

Abstract

We simplify the proof of the optimality of the Shortest Lower-Bound First (SLF) algorithm, introduced by Gupta, Kaplan, Lindermayr, Schlöter, and Yingchareonthawornchai [FOCS'25], for minimizing the total flow time in the -clairvoyant setting.
Paper Structure (10 sections, 11 theorems, 10 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Frozen Jobs instead of Early Arrivals.
  4. Fixed-Length Prefix Bounds instead of Valid Assignments.
  5. The Shortest-Lower-Bound-First Algorithm
  6. Analysis
  7. Terminology.
  8. Fast Forward Lemma
  9. Suffix Carving Lemma
  10. Proof of \ref{['lem:volume-bound']}

Key Result

Theorem 1.1

For any constant $\varepsilon \in (0,1]$, there is a deterministic $\varepsilon$-clairvoyant algorithm SLF that is ${\left\lceil 1/\varepsilon \right\rceil}$-competitive for the objective of minimizing the total flow time on a single machine.

Figures (1)

  • Figure 1: Schedule of SLF and corresponding inductive proof of \ref{['lem:volume-bound']}. The arrows indicate the application of either the Fast Forward Lemma (FF) or the Suffix Carving Lemma (SC) at the respective time intervals, and the corresponding case in the proof. Here, $s_j$ denotes the point in time when job $j$ becomes known, $r_j$ denotes the arrival time of job $j$, and $C_j$ denotes the completion time of job $j$.

Theorems & Definitions (25)

  • Theorem 1.1: GuptaKLSY25
  • Definition 2.1: Known/Unknown Jobs
  • Lemma 3.1
  • Lemma 3.2
  • proof : Proof of \ref{['lem:local-competitiveness']}
  • Definition 3.3: Frozen jobs
  • Definition 3.4: Leader
  • Lemma 3.5
  • proof
  • Lemma 3.6: Fast Forward Lemma
  • ...and 15 more