Faster Deterministic Streaming Vertex Coloring

Abstract

Graph coloring is a fundamental problem in computer science. In the semi-streaming model, an input graph G on n vertices and maximum degree Δ is presented as a stream of edges, and the goal is to compute a vertex coloring using a small number of colors while storing only Õ(n) bits of memory.
Recent work has revealed an exponential separation between randomized and deterministic approaches in this setting: while randomized algorithms can achieve a (Δ+1)-coloring in a single pass [Assadi, Chen, and Khanna, 2019], any single-pass deterministic algorithm requires exp(Δ^Ω(1)) colors [Assadi, Chen, and Sun, 2022]. Consequently, deterministic algorithms that use few colors must necessarily make multiple passes over the stream. Prior to this work, the best known deterministic trade-offs were: an O(Δ²)-coloring in 2 passes, an O(Δ)-coloring in O(log Δ) passes [Assadi, Chen, and Sun, 2022], and a (Δ+1)-coloring in O(log Δ ⋅ log log Δ) passes [Assadi, Chakrabarti, Ghosh, and Stoeckl, 2023]. It remained open whether better trade-offs - particularly with sub-logarithmic pass complexity and linear-in-Δ palette size - were achievable.
In this paper, we present a new deterministic semi-streaming algorithm that computes an O(Δ)-coloring in O(√{log Δ}) passes. This is the first deterministic streaming algorithm to achieve a coloring with palette size linear-in-Δ using sublogarithmic-in-Δ passes.