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Online Edge Coloring is (Nearly) as Easy as Offline

Joakim Blikstad, Ola Svensson, Radu Vintan, David Wajc

TL;DR

This work generalizes this result to obtain online counterparts of the list edge coloring result of Kahn (J. Comb. Theory. A’96) and of the recent “local” edge coloring result of Christiansen (STOC’23).

Abstract

The classic theorem of Vizing (Diskret. Analiz.'64) asserts that any graph of maximum degree $Δ$ can be edge colored (offline) using no more than $Δ+1$ colors (with $Δ$ being a trivial lower bound). In the online setting, Bar-Noy, Motwani and Naor (IPL'92) conjectured that a $(1+o(1))Δ$-edge-coloring can be computed online in $n$-vertex graphs of maximum degree $Δ=ω(\log n)$. Numerous algorithms made progress on this question, using a higher number of colors or assuming restricted arrival models, such as random-order edge arrivals or vertex arrivals (e.g., AGKM FOCS'03, BMM SODA'10, CPW FOCS'19, BGW SODA'21, KLSST STOC'22). In this work, we resolve this longstanding conjecture in the affirmative in the most general setting of adversarial edge arrivals. We further generalize this result to obtain online counterparts of the list edge coloring result of Kahn (J. Comb. Theory. A'96) and of the recent "local" edge coloring result of Christiansen (STOC'23).

Online Edge Coloring is (Nearly) as Easy as Offline

TL;DR

This work generalizes this result to obtain online counterparts of the list edge coloring result of Kahn (J. Comb. Theory. A’96) and of the recent “local” edge coloring result of Christiansen (STOC’23).

Abstract

The classic theorem of Vizing (Diskret. Analiz.'64) asserts that any graph of maximum degree can be edge colored (offline) using no more than colors (with being a trivial lower bound). In the online setting, Bar-Noy, Motwani and Naor (IPL'92) conjectured that a -edge-coloring can be computed online in -vertex graphs of maximum degree . Numerous algorithms made progress on this question, using a higher number of colors or assuming restricted arrival models, such as random-order edge arrivals or vertex arrivals (e.g., AGKM FOCS'03, BMM SODA'10, CPW FOCS'19, BGW SODA'21, KLSST STOC'22). In this work, we resolve this longstanding conjecture in the affirmative in the most general setting of adversarial edge arrivals. We further generalize this result to obtain online counterparts of the list edge coloring result of Kahn (J. Comb. Theory. A'96) and of the recent "local" edge coloring result of Christiansen (STOC'23).
Paper Structure (32 sections, 37 theorems, 106 equations, 3 figures, 5 algorithms)

This paper contains 32 sections, 37 theorems, 106 equations, 3 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Main Result.
  3. Techniques overview.
  4. Secondary Results and Extensions.
  5. Related Work
  6. Preliminaries
  7. Problem definition and notation.
  8. Martingales.
  9. Online Matching Algorithm
  10. Our First Matching Algorithm
  11. The Analysis-Friendly Matching Algorithm
  12. Analysis: Intuition and Overview.
  13. Analysis of the Online Matching Algorithm
  14. Choice of $q$.
  15. A Sufficient Condition for 1/(Δ+q) Matching Probability
  16. ...and 17 more sections

Key Result

theorem 1.2

There exists an online algorithm that, on $n$-vertex graphs with known maximum degree $\Delta=\omega(\log n)$, outputs a $(1+o(1))\Delta$-edge-coloring with high probability.

Figures (3)

  • Figure 1: An example of the neighborhood of $e_t = (u,v)$ with $k = 7$.
  • Figure 2: Example where \ref{['alg:natural-alg']} might be undefined.
  • Figure 3: The configuration described in \ref{['lemma:main_ineq']}, i.e., the $2$-hop neighborhood of $v$. The (at most) $\Delta^2$ edges drawn correspond to the only non-trivial steps of the martingale.

Theorems & Definitions (72)

  • conjecture 1.1: bar1992greedy
  • theorem 1.2: See exact bounds in \ref{['thm:ec-body']}
  • theorem 1.3
  • theorem 1.4: See exact bounds in \ref{['thm:list_edge_coloring']}
  • theorem 1.5: See exact bounds in \ref{['thm:local_edge_coloring_strenghtened']}
  • theorem 1.6: See exact bounds in \ref{['thm:ronding_matchings_theorem']}
  • lemma 2.1: Reduction (cohen2019tightsaberi2021greedy)
  • definition 2.2: Martingale
  • lemma 2.3: Freedman's Inequality freedman1975tail; see also freedman, habib1998probabilistic
  • proof
  • ...and 62 more