A linear-time algorithm for $(1+ε)Δ$-edge-coloring

We present a randomized algorithm that, given $\epsilon > 0$, outputs a proper $(1+\epsilon)\Delta$-edge-coloring of an $m$-edge simple graph $G$ of maximum degree $\Delta \geq 1/\epsilon$ in $O(m\,\log(1/\epsilon)/\epsilon^4)$ time. For constant $\epsilon$, this is the first linear-time algorithm for this problem without any restrictions on $\Delta$ other than the necessary bound $\Delta \geq 1/\epsilon$. The best previous result in this direction, very recently obtained by Assadi, gives a randomized algorithm with expected running time $O(m \, \log(1/\epsilon))$ under the assumption $\Delta \gg \log n/\epsilon$; removing the lower bound on $\Delta$ was explicitly mentioned as a challenging open problem by Bhattacharya, Costa, Panski, and Solomon. Indeed, even for edge-coloring with $2\Delta - 1$ colors (i.e., meeting the "greedy" bound), no linear-time algorithm covering the full range of $\Delta$ has been known until now. Additionally, when $\epsilon = 1/\Delta$, our result yields an $O(m\,\Delta^4\log \Delta)$-time algorithm for $(\Delta+1)$-edge-coloring, improving the bound $O(m\, \Delta^{17})$ from the authors' earlier work.