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Randomized Greedy Online Edge Coloring Succeeds for Dense and Randomly-Ordered Graphs

Aditi Dudeja, Rashmika Goswami, Michael Saks

TL;DR

This work shows that the random greedy algorithm simply chooses a legal color uniformly at random for each edge upon arrival in two contexts and implies the existence of a deterministic edge coloring algorithm which $(1+\epsilon)\Delta$ edge colors a dense graph.

Abstract

Vizing's theorem states that any graph of maximum degree $Δ$ can be properly edge colored with at most $Δ+1$ colors. In the online setting, it has been a matter of interest to find an algorithm that can properly edge color any graph on $n$ vertices with maximum degree $Δ= ω(\log n)$ using at most $(1+o(1))Δ$ colors. Here we study the naïve random greedy algorithm, which simply chooses a legal color uniformly at random for each edge upon arrival. We show that this algorithm can $(1+ε)Δ$-color the graph for arbitrary $ε$ in two contexts: first, if the edges arrive in a uniformly random order, and second, if the edges arrive in an adversarial order but the graph is sufficiently dense, i.e., $n = O(Δ)$. Prior to this work, the random greedy algorithm was only known to succeed in trees. Our second result is applicable even when the adversary is adaptive, and therefore implies the existence of a deterministic edge coloring algorithm which $(1+ε)Δ$ edge colors a dense graph. Prior to this, the best known deterministic algorithm for this problem was the simple greedy algorithm which utilized $2Δ-1$ colors.

Randomized Greedy Online Edge Coloring Succeeds for Dense and Randomly-Ordered Graphs

TL;DR

This work shows that the random greedy algorithm simply chooses a legal color uniformly at random for each edge upon arrival in two contexts and implies the existence of a deterministic edge coloring algorithm which edge colors a dense graph.

Abstract

Vizing's theorem states that any graph of maximum degree can be properly edge colored with at most colors. In the online setting, it has been a matter of interest to find an algorithm that can properly edge color any graph on vertices with maximum degree using at most colors. Here we study the naïve random greedy algorithm, which simply chooses a legal color uniformly at random for each edge upon arrival. We show that this algorithm can -color the graph for arbitrary in two contexts: first, if the edges arrive in a uniformly random order, and second, if the edges arrive in an adversarial order but the graph is sufficiently dense, i.e., . Prior to this work, the random greedy algorithm was only known to succeed in trees. Our second result is applicable even when the adversary is adaptive, and therefore implies the existence of a deterministic edge coloring algorithm which edge colors a dense graph. Prior to this, the best known deterministic algorithm for this problem was the simple greedy algorithm which utilized colors.
Paper Structure (27 sections, 23 theorems, 97 equations, 1 figure)

This paper contains 27 sections, 23 theorems, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work On Edge Coloring.
  3. Edge Coloring in Other Computational Models.
  4. Our Approach
  5. Comparison to Prior Work.
  6. Paper Outline.
  7. Preliminaries
  8. Online Coloring
  9. The Algorithm $\mathcal{A}'$
  10. Algorithm Description
  11. The probability space
  12. Notation.
  13. Proof Idea
  14. Technical parameters
  15. Properties of the phase partitions
  16. ...and 12 more sections

Key Result

Theorem 1

( Random order case) Let $\epsilon>0$ be a constant. The algorithm $\mathcal{A}$, when given any simple graph $G$ of maximum degree $\Delta=\omega(\log n)$, whose edges are presented in a uniformly random order, edge colors $G$ with $(1+\epsilon)\Delta$ colors with high probability.

Figures (1)

  • Figure 1: A simple example to illustrate that if $\left|\Gamma\right|=\Delta+o(\sqrt{\Delta})$, then $\mathcal{A}$ likely fails.

Theorems & Definitions (56)

  • Theorem 1: Informal version of \ref{['thm:randomordermain']}
  • Theorem 2: Informal version of \ref{['thm:main']}
  • Corollary 3
  • Theorem 4
  • proof
  • Theorem 4
  • Theorem 4
  • Definition 5: Algorithm $\mathcal{A}'$
  • Definition 6: Vertex-Phase and Edge Arrival Ordering, $\prec$
  • Definition 7: Technical parameters
  • ...and 46 more