Streaming Edge Coloring with Subquadratic Palette Size
Shiri Chechik, Doron Mukhtar, Tianyi Zhang
TL;DR
This paper tackles one-pass edge-coloring in the W-streaming model under adversarial input with near-linear space. It introduces a phase-based, randomized algorithm that partitions the stream and uses multiple palettes alongside per-vertex random offsets to color edges while pushing uncolorable edges to a recursive virtual stream. The main contribution is a subquadratic palette size bound: $\tilde{O}(Δ^{1.5})$ colors in $\tilde{O}(n)$ space in expectation, applicable to adversarial order streams. The approach extends to unknown $Δ$ and can achieve high-probability guarantees, advancing the longstanding open problem of achieving $O(Δ)$-coloring in this model.
Abstract
In this paper, we study the problem of computing an edge-coloring in the (one-pass) W-streaming model. In this setting, the edges of an $n$-node graph arrive in an arbitrary order to a machine with a relatively small space, and the goal is to design an algorithm that outputs, as a stream, a proper coloring of the edges using the fewest possible number of colors. Behnezhad et al. [Behnezhad et al., 2019] devised the first non-trivial algorithm for this problem, which computes in $\tilde{O}(n)$ space a proper $O(Δ^2)$-coloring w.h.p. (here $Δ$ is the maximum degree of the graph). Subsequent papers improved upon this result, where latest of them [Ansari et al., 2022] shows that it is possible to deterministically compute an $O(Δ^2/s)$-coloring in $O(ns)$ space. However, none of the improvements, succeeded in reducing the number of colors to $O(Δ^{2-ε})$ while keeping the same space bound of $\tilde{O}(n)$. In particular, no progress was made on the question of whether computing an $O(Δ)$-coloring is possible with roughly $O(n)$ space, which was stated in [Behnezhad et al., 2019] to be a major open problem. In this paper we bypass the quadratic bound by presenting a new randomized $\tilde{O}(n)$-space algorithm that uses $\tilde{O}(Δ^{1.5})$ colors.