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Fully Dynamic (Δ+1) Coloring Against Adaptive Adversaries

Soheil Behnezhad, Rajmohan Rajaraman, Omer Wasim

TL;DR

This work tackles maintaining a $(\Delta+1)$-coloring in fully dynamic graphs under adaptive adversaries, a challenging setting where updates depend on the current output. The authors develop a sparse-dense decomposition that partitions vertices into a sparse set $V_S$ and dense almost-cliques in $V_D$, enabling targeted recoloring strategies. They introduce a phase-based approach where a fixed decomposition is used for $t=\Theta(\varepsilon^2\Delta)$ updates, then refreshed, achieving a sublinear amortized update time of $\widetilde{O}(n^{8/9})$ with high probability. The method combines dynamic non-edge matchings, Step I/II colorings for dense parts, and a robust maintenance framework for the sparse-dense structure, breaking the previous linear-time barrier against adaptive adversaries and offering insights for sublinear dynamic graph algorithms with adversarial updates.

Abstract

Over the years, there has been extensive work on fully dynamic algorithms for classic graph problems that admit greedy solutions. Examples include $(Δ+1)$ vertex coloring, maximal independent set, and maximal matching. For all three problems, there are randomized algorithms that maintain a valid solution after each edge insertion or deletion to the $n$-vertex graph by spending $\polylog n$ time, provided that the adversary is oblivious. However, none of these algorithms work against adaptive adversaries whose updates may depend on the output of the algorithm. In fact, even breaking the trivial bound of $O(n)$ against adaptive adversaries remains open for all three problems. For instance, in the case of $(Δ+1)$ vertex coloring, the main challenge is that an adaptive adversary can keep inserting edges between vertices of the same color, necessitating a recoloring of one of the endpoints. The trivial algorithm would simply scan all neighbors of one endpoint to find a new available color (which always exists) in $O(n)$ time. In this paper, we break this linear barrier for the $(Δ+1)$ vertex coloring problem. Our algorithm is randomized, and maintains a valid $(Δ+1)$ vertex coloring after each edge update by spending $\widetilde{O}(n^{8/9})$ time with high probability.

Fully Dynamic (Δ+1) Coloring Against Adaptive Adversaries

TL;DR

This work tackles maintaining a -coloring in fully dynamic graphs under adaptive adversaries, a challenging setting where updates depend on the current output. The authors develop a sparse-dense decomposition that partitions vertices into a sparse set and dense almost-cliques in , enabling targeted recoloring strategies. They introduce a phase-based approach where a fixed decomposition is used for updates, then refreshed, achieving a sublinear amortized update time of with high probability. The method combines dynamic non-edge matchings, Step I/II colorings for dense parts, and a robust maintenance framework for the sparse-dense structure, breaking the previous linear-time barrier against adaptive adversaries and offering insights for sublinear dynamic graph algorithms with adversarial updates.

Abstract

Over the years, there has been extensive work on fully dynamic algorithms for classic graph problems that admit greedy solutions. Examples include vertex coloring, maximal independent set, and maximal matching. For all three problems, there are randomized algorithms that maintain a valid solution after each edge insertion or deletion to the -vertex graph by spending time, provided that the adversary is oblivious. However, none of these algorithms work against adaptive adversaries whose updates may depend on the output of the algorithm. In fact, even breaking the trivial bound of against adaptive adversaries remains open for all three problems. For instance, in the case of vertex coloring, the main challenge is that an adaptive adversary can keep inserting edges between vertices of the same color, necessitating a recoloring of one of the endpoints. The trivial algorithm would simply scan all neighbors of one endpoint to find a new available color (which always exists) in time. In this paper, we break this linear barrier for the vertex coloring problem. Our algorithm is randomized, and maintains a valid vertex coloring after each edge update by spending time with high probability.

Paper Structure

This paper contains 42 sections, 49 theorems, 19 equations, 4 figures, 17 algorithms.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Background on Sparse-Dense Decompositions for D+1 Vertex Coloring
  4. Our Dynamic Coloring: Basic Invariants and Parameters
  5. Coloring Sparse Vertices Dynamically
  6. Coloring Dense Vertices Dynamically
  7. Putting Everything Together
  8. Preliminaries
  9. Organization of the Paper
  10. A Fully Dynamic Sparse-Dense Decomposition
  11. A Fully Dynamic D+1-Coloring Algorithm
  12. Working with a Fixed Decomposition for e2D updates
  13. An Algorithm for Sparse Vertices
  14. Recoloring Sparse Vertices at the Beginning of a Phase
  15. Recoloring Sparse Vertices During a Phase
  16. ...and 27 more sections

Key Result

Theorem 1

There exists a fully dynamic algorithm for $(\Delta+1)$-coloring taking amortized update time of $\widetilde{O}(n^{8/9})$ w.h.p., against an adaptive adversary.

Figures (4)

  • Figure 2.1: An example of sparse-dense decomposition.
  • Figure 5.1: The length five augmenting path we find for coloring a vertex $v$ in small almost-cliques.
  • Figure :
  • Figure :

Theorems & Definitions (103)

  • Theorem 1
  • Definition 1: $\varepsilon$-friend edges hss
  • Definition 2: $\varepsilon$-dense vertices hss
  • Lemma 1: hss
  • Definition 3
  • Theorem 2
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • ...and 93 more