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Deterministic Dynamic Edge Colouring

Aleksander B. G. Christiansen

TL;DR

We present a deterministic dynamic algorithm for maintaining a $(1+\varepsilon)\Delta$-edge-colouring of a dynamic graph with amortised update time $2^{\tilde{O}_{\log \varepsilon^{-1}}(\sqrt{\log n})}$, improving on the greedy threshold of $2\Delta-1$ without randomness. The approach combines a dynamic $t$-splitter to partition edges into low-degree subgraphs and a sophisticated, deterministic treatment of low-degree graphs using (non-overlapping) multi-step Vizing chains and carefully maintained data-structures. A shallow hierarchy of dynamic splitters reduces degrees down to poly$(\log n, \varepsilon^{-1})$, enabling efficient global recombination into a single coloured graph; this hierarchy is argued to have sub-polynomial recourse and update-time. The paper also introduces a robust framework of path-stepping processes and a bichromatic skeleton to support efficient, deterministic recolouring, with potential independent applicability beyond edge-colouring. Taken together, these results close the gap with randomized dynamic algorithms by providing deterministic guarantees with near-optimal colour usage and sub-polynomial update-time in a broad parameter regime.

Abstract

Given a dynamic graph $G$ with $n$ vertices and $m$ edges subject to insertion an deletions of edges, we show how to maintain a $(1+\varepsilon)Δ$-edge-colouring of $G$ without the use of randomisation. More specifically, we show a deterministic dynamic algorithm with an amortised update time of $2^{\tilde{O}_{\log \varepsilon^{-1}}(\sqrt{\log n})}$ using $(1+\varepsilon)Δ$ colours. If $\varepsilon^{-1} \in 2^{O(\log^{0.49} n)}$, then our update time is sub-polynomial in $n$. While there exists randomised algorithms maintaining colourings with the same number of colours [Christiansen STOC'23, Duan, He, Zhang SODA'19, Bhattacarya, Costa, Panski, Solomon SODA'24] in polylogarithmic and even constant update time, this is the first deterministic algorithm to go below the greedy threshold of $2Δ-1$ colours for all input graphs. On the way to our main result, we show how to dynamically maintain a shallow hierarchy of degree-splitters with both recourse and update time in $n^{o(1)}$. We believe that this algorithm might be of independent interest.

Deterministic Dynamic Edge Colouring

TL;DR

We present a deterministic dynamic algorithm for maintaining a -edge-colouring of a dynamic graph with amortised update time , improving on the greedy threshold of without randomness. The approach combines a dynamic -splitter to partition edges into low-degree subgraphs and a sophisticated, deterministic treatment of low-degree graphs using (non-overlapping) multi-step Vizing chains and carefully maintained data-structures. A shallow hierarchy of dynamic splitters reduces degrees down to poly, enabling efficient global recombination into a single coloured graph; this hierarchy is argued to have sub-polynomial recourse and update-time. The paper also introduces a robust framework of path-stepping processes and a bichromatic skeleton to support efficient, deterministic recolouring, with potential independent applicability beyond edge-colouring. Taken together, these results close the gap with randomized dynamic algorithms by providing deterministic guarantees with near-optimal colour usage and sub-polynomial update-time in a broad parameter regime.

Abstract

Given a dynamic graph with vertices and edges subject to insertion an deletions of edges, we show how to maintain a -edge-colouring of without the use of randomisation. More specifically, we show a deterministic dynamic algorithm with an amortised update time of using colours. If , then our update time is sub-polynomial in . While there exists randomised algorithms maintaining colourings with the same number of colours [Christiansen STOC'23, Duan, He, Zhang SODA'19, Bhattacarya, Costa, Panski, Solomon SODA'24] in polylogarithmic and even constant update time, this is the first deterministic algorithm to go below the greedy threshold of colours for all input graphs. On the way to our main result, we show how to dynamically maintain a shallow hierarchy of degree-splitters with both recourse and update time in . We believe that this algorithm might be of independent interest.
Paper Structure (50 sections, 49 theorems, 69 equations, 2 figures)

This paper contains 50 sections, 49 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction & Related Work
  2. Main technical challenges
  3. Further Related Work
  4. Results
  5. Informal Overview
  6. The overall approach
  7. Dynamic $t$-splitters
  8. $(\Delta_{\max}+1)$-edge-colouring of low degree graphs
  9. Outline of the paper
  10. Preliminaries & Notation
  11. Chains and shifts
  12. Algorithms for constructing and extending Vizing chains
  13. Case 1:
  14. Case 2:
  15. Case 3:
  16. ...and 35 more sections

Key Result

Theorem 1.1

Let $G$ be a dynamic graph and $\varepsilon^{-1} \leq 2^{O(\log^{0.49} n)}$ a given parameter. Then, we can maintain a proper $((1+\varepsilon)\Delta)$-edge-colouring of $G$ in $n^{o(1)}$ amortised update time per operation.

Figures (2)

  • Figure 2.1: An augmenting Vizing chain is shifted on the left, and a truncated Vizing chain is shifted on the right. The above illustration is originally due to Christiansen Christiansen.
  • Figure 2.2: On the left, we see a non-overlapping multi-step Vizing chain. On the right, we see a multi-step Vizing chain overlapping in a single edge. The above illustration is originally due to Christiansen Christiansen.

Theorems & Definitions (106)

  • Theorem 1.1: Informal
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4: 2BERNSHTEYNBernDhawanvizing1964estimate
  • Proof 1
  • ...and 96 more