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3.415-Approximation for Coflow Scheduling via Iterated Rounding

Lars Rohwedder, Leander Schnaars

TL;DR

This work significantly advances Coflow Scheduling by beating the classic $4$- and $5$-approximation bounds through a novel iterated LP rounding and edge-allocation framework. It couples LP-guided deadline selection with two complementary edge-allocation schemes, Greedy and Beck-Fiala-based iterated rounding, and unifies them via a general combining guarantees framework to produce a $140/41$-approximation without release dates and a $4.36$-approximation with release dates. An asymptotic analysis yields a near-optimal $2+\epsilon$-approximation in regimes where most coflows finish late, underscoring the method's strength in practical large-scale settings. The work also provides rigorous appendix material on LP integrality gaps, NP-hardness, high multiplicities, and potential refinements, with implications for online extensions via existing online-to-offline reductions.

Abstract

We provide an algorithm giving a $\frac{140}{41}$($<3.415$)-approximation for Coflow Scheduling and a $4.36$-approximation for Coflow Scheduling with release dates. This improves upon the best known $4$- and respectively $5$-approximations and addresses an open question posed by Agarwal, Rajakrishnan, Narayan, Agarwal, Shmoys, and Vahdat [Aga+18], Fukunaga [Fuk22], and others. We additionally show that in an asymptotic setting, the algorithm achieves a ($2+ε$)-approximation, which is essentially optimal under $\mathbb{P}\neq\mathbb{NP}$. The improvements are achieved using a novel edge allocation scheme using iterated LP rounding together with a framework which enables establishing strong bounds for combinations of several edge allocation algorithms.

3.415-Approximation for Coflow Scheduling via Iterated Rounding

TL;DR

This work significantly advances Coflow Scheduling by beating the classic - and -approximation bounds through a novel iterated LP rounding and edge-allocation framework. It couples LP-guided deadline selection with two complementary edge-allocation schemes, Greedy and Beck-Fiala-based iterated rounding, and unifies them via a general combining guarantees framework to produce a -approximation without release dates and a -approximation with release dates. An asymptotic analysis yields a near-optimal -approximation in regimes where most coflows finish late, underscoring the method's strength in practical large-scale settings. The work also provides rigorous appendix material on LP integrality gaps, NP-hardness, high multiplicities, and potential refinements, with implications for online extensions via existing online-to-offline reductions.

Abstract

We provide an algorithm giving a ()-approximation for Coflow Scheduling and a -approximation for Coflow Scheduling with release dates. This improves upon the best known - and respectively -approximations and addresses an open question posed by Agarwal, Rajakrishnan, Narayan, Agarwal, Shmoys, and Vahdat [Aga+18], Fukunaga [Fuk22], and others. We additionally show that in an asymptotic setting, the algorithm achieves a ()-approximation, which is essentially optimal under . The improvements are achieved using a novel edge allocation scheme using iterated LP rounding together with a framework which enables establishing strong bounds for combinations of several edge allocation algorithms.

Paper Structure

This paper contains 20 sections, 31 theorems, 53 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Organization
  4. Preliminaries
  5. Algorithmic Framework
  6. Coflow Deadlines
  7. Integral Edge Assignments with Guarantees
  8. Combining Algorithmic Guarantees
  9. Main Theorem
  10. Integral Edge Assignments with Guarantees
  11. Greedy Scheduling
  12. Iterated Rounding using Beck-Fiala
  13. Asymptotic $2+\epsilon$ Approximation
  14. Appendix
  15. Allocation LP Integrality and Complexity
  16. ...and 5 more sections

Key Result

Theorem 1

There is a polynomial time algorithm achieving a $\frac{140}{41} (<3.415)$-approximation for Coflow Scheduling without release dates.

Figures (1)

  • Figure 1: The first picture shows a feasible fractional allocation, where dashed lines indicate values of $\tfrac{1}{2}$. The second and third picture show the two allocations for $E_1 \cup E_2 \cup E_3 \cup E_4^1$, with the relevant $V$ vertices which are blocked by gadget edges replaced by filled vertices for easier readability.

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: König1916
  • Lemma 4: im19
  • Lemma 4
  • Theorem 5
  • Lemma 5
  • proof
  • Theorem 5
  • ...and 41 more