$(Δ+ 1)$ Vertex Coloring in $O(n)$ Communication

TL;DR

This work resolves the randomized communication complexity of $(Δ+1)$-coloring in the two-party model by delivering an $O(n)$-bit protocol that outputs a proper coloring (in expectation) and a matching $oldsymbol{Ω(n)}$ lower bound for constant-error protocols. The key ideas combine a non-deterministic upper bound based on a large catalog of colorings with a novel randomized palette-sparsification scheme and a per-vertex $k$-Slack-Int subroutine, reducing the per-vertex communication to $O(\\log^{2}(Δ/k))$ and enabling a global $O(n)$ bound. The approach also yields high-probability guarantees that depend on the relationship between $oldsymbol{Δ}$ and $oldsymbol{n}$, including $O(n)$ w.h.p. for small $oldsymbol{Δ}$ and $O(n \,\log^{*} Δ)$ w.h.p. for large $oldsymbol{Δ}$. The results close the gap between trivial linear lower bounds and nearly-linear upper bounds in the randomized regime, and open several directions for deterministic bounds and extensions to related coloring variants.

Abstract

We study the communication complexity of $(Δ+ 1)$ vertex coloring, where the edges of an $n$-vertex graph of maximum degree $Δ$ are partitioned between two players. We provide a randomized protocol which uses $O(n)$ bits of communication and ends with both players knowing the coloring. Combining this with a folklore $Ω(n)$ lower bound, this settles the randomized communication complexity of $(Δ+ 1)$-coloring up to constant factors.

Figures (1)

• Figure 1: The gadget encoding a single bit. The dashed red edges are present when the bit is $0$, and the dotted blue edges are present when the bit is $1$. The solid black edges are always present.

Theorems & Definitions (32)

• Theorem 1
• Corollary 1.1
• Proposition 2.1
• proof
• Proposition 2.2: Chernoff Bound
• Proposition 2.3: Hoeffding Bound Hoeffding94
• Proposition 2.4: Bounded Differences DubhashiP09
• Theorem 2
• Lemma 3.1
• proof