Coloring in Graph Streams via Deterministic and Adversarially Robust Algorithms

TL;DR

This paper advances graph coloring in streaming models by developing deterministic and adversarially robust algorithms that improve color bounds and pass complexity, approaching theoretical optimality.

Contribution

It introduces new deterministic and adversarially robust algorithms for graph coloring in semi-streaming, reducing color bounds and passes compared to prior work.

Findings

Deterministic semi-streaming algorithm achieves $( ext{Δ}+1)$-coloring with $O( ext{log} ext{Δ} ext{log} ext{log} ext{Δ})$ passes.

Adversarially robust algorithms improve color bounds from $O( ext{Δ}^3)$ to $O( ext{Δ}^{5/2})$ and provide space-color tradeoffs.

Algorithms approach theoretical bounds for graph coloring in streaming and adversarial settings.

Abstract

In recent years, there has been a growing interest in solving various graph coloring problems in the streaming model. The initial algorithms in this line of work are all crucially randomized, raising natural questions about how important a role randomization plays in streaming graph coloring. A couple of very recent works have made progress on this question: they prove that deterministic or even adversarially robust coloring algorithms (that work on streams whose updates may depend on the algorithm's past outputs) are considerably weaker than standard randomized ones. However, there is still a significant gap between the upper and lower bounds for the number of colors needed (as a function of the maximum degree $\Delta$) for robust coloring and multipass deterministic coloring. We contribute to this line of work by proving the following results. In the deterministic semi-streaming (i.e., $O(n \cdot \text{polylog } n)$ space) regime, we present an algorithm that achieves a combinatorially optimal $(\Delta+1)$-coloring using $O(\log{\Delta} \log\log{\Delta})$ passes. This improves upon the prior $O(\Delta)$-coloring algorithm of Assadi, Chen, and Sun (STOC 2022) at the cost of only an $O(\log\log{\Delta})$ factor in the number of passes. In the adversarially robust semi-streaming regime, we design an $O(\Delta^{5/2})$-coloring algorithm that improves upon the previously best $O(\Delta^{3})$-coloring algorithm of Chakrabarti, Ghosh, and Stoeckl (ITCS 2022). Further, we obtain a smooth colors/space tradeoff that improves upon another algorithm of the said work: whereas their algorithm uses $O(\Delta^2)$ colors and $O(n\Delta^{1/2})$ space, ours, in particular, achieves (i)~$O(\Delta^2)$ colors in $O(n\Delta^{1/3})$ space, and (ii)~$O(\Delta^{7/4})$ colors in $O(n\Delta^{1/2})$ space.