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Arboricity-Dependent Algorithms for Edge Coloring

Sayan Bhattacharya, Martín Costa, Nadav Panski, Shay Solomon

TL;DR

The paper studies dynamic edge coloring in graphs with bounded arboricity, showing a deterministic algorithm can maintain a $(Δ + O(α))$-edge coloring by exploiting a peeling-based decomposition and a discretized decomposition system to handle changing arboricity. The approach dynamically maintains local coloring constraints via ExtendColoring, with provable amortized bounds on update time and recolorings, and is adaptive to time-varying $Δ_t$ and $α_t$. For fixed arboricity, a warmup yields efficient recoloring with polylogarithmic update time; in general, the full algorithm achieves a $(Δ + O(α))$ coloring with near-constant recourse and polylogarithmic update time, particularly when $α leq Δ^{1-μ}$ for constant μ>0. These results advance dynamic graph algorithms by providing near-optimal additive colorings on sparse graphs (eg, planar, bounded-genus) under continual updates, with robust performance as graph parameters evolve.

Abstract

The problem of edge coloring has been extensively studied over the years. Recently, this problem has received significant attention in the dynamic setting, where we are given a dynamic graph evolving via a sequence of edge insertions and deletions and our objective is to maintain an edge coloring of the graph. Currently, it is not known whether it is possible to maintain a $(Δ+ O(Δ^{1 - μ}))$-edge coloring in $\tilde{O}(1)$ update time, for any constant $μ> 0$, where $Δ$ is the maximum degree of the graph. In this paper, we show how to efficiently maintain a $(Δ+ O(α))$-edge coloring in $\tilde O(1)$ amortized update time, where $α$ is the arboricty of the graph. Thus, we answer this question in the affirmative for graphs of sufficiently small arboricity.

Arboricity-Dependent Algorithms for Edge Coloring

TL;DR

The paper studies dynamic edge coloring in graphs with bounded arboricity, showing a deterministic algorithm can maintain a -edge coloring by exploiting a peeling-based decomposition and a discretized decomposition system to handle changing arboricity. The approach dynamically maintains local coloring constraints via ExtendColoring, with provable amortized bounds on update time and recolorings, and is adaptive to time-varying and . For fixed arboricity, a warmup yields efficient recoloring with polylogarithmic update time; in general, the full algorithm achieves a coloring with near-constant recourse and polylogarithmic update time, particularly when for constant μ>0. These results advance dynamic graph algorithms by providing near-optimal additive colorings on sparse graphs (eg, planar, bounded-genus) under continual updates, with robust performance as graph parameters evolve.

Abstract

The problem of edge coloring has been extensively studied over the years. Recently, this problem has received significant attention in the dynamic setting, where we are given a dynamic graph evolving via a sequence of edge insertions and deletions and our objective is to maintain an edge coloring of the graph. Currently, it is not known whether it is possible to maintain a -edge coloring in update time, for any constant , where is the maximum degree of the graph. In this paper, we show how to efficiently maintain a -edge coloring in amortized update time, where is the arboricty of the graph. Thus, we answer this question in the affirmative for graphs of sufficiently small arboricity.
Paper Structure (25 sections, 30 theorems, 2 equations, 10 algorithms)

This paper contains 25 sections, 30 theorems, 2 equations, 10 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Our Techniques
  4. Handling the scenario where $\Delta$ and $\alpha$ change over time
  5. Roadmap
  6. Preliminaries
  7. The Dynamic Setting
  8. Notation
  9. Graph Decompositions
  10. Graph Decomposition Systems
  11. A Warmup Dynamic Algorithm (for Fixed $\alpha$)
  12. Algorithm Description
  13. Analysis of the Warmup Algorithm
  14. The Dynamic Algorithm
  15. Algorithm Description
  16. ...and 10 more sections

Key Result

Theorem 1

There is a deterministic dynamic algorithm for maintaining a $(\Delta + (4+\epsilon)\alpha)$-edge coloring of an input dynamic graph with maximum degree $\Delta$ and arboricity $\alpha$, with $O(\log^6n /\epsilon^6)$ amortized update time and $O(\log^4 n / \epsilon^5)$ amortized recourse.A dynamic a

Theorems & Definitions (34)

  • Theorem 1
  • Corollary 2
  • Definition 3
  • Lemma 4: BhattacharyaHNT15
  • Proposition 5: BhattacharyaHNT15
  • Definition 6
  • Proposition 7
  • Theorem 8
  • Lemma 9
  • Lemma 10
  • ...and 24 more