English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the problem of maintaining a $(1+\epsilon)\Delta$-edge coloringin a dynamic graph $G$ with $n$ nodes and maximum degree at most $\Delta$. Thestate-of-the-art update time is $O_\epsilon(\text{polylog}(n))$, by Duan, Heand Zhang [SODA'19] and by Christiansen [STOC'23], and more precisely $O(\log^7n/\epsilon^2)$, where $\Delta = \Omega(\log^2 n / \epsilon^2)$.  The following natural question arises: What is the best possible update timeof an algorithm for this task? More specifically, \textbf{ can we bring it allthe way down to some constant} (for constant $\epsilon$)? This questioncoincides with the \emph{static} time barrier for the problem: Even for$(2\Delta-1)$-coloring, there is only a naive $O(m \log \Delta)$-timealgorithm.  We answer this fundamental question in the affirmative, by presenting adynamic $(1+\epsilon)\Delta$-edge coloring algorithm with $O(\log^4(1/\epsilon)/\epsilon^9)$ update time, provided $\Delta =\Omega_\epsilon(\text{polylog}(n))$. As a corollary, we also get the firstlinear time (for constant $\epsilon$) \emph{static} algorithm for$(1+\epsilon)\Delta$-edge coloring; in particular, we achieve a running time of$O(m \log (1/\epsilon)/\epsilon^2)$.  We obtain our results by carefully combining a variant of the \textsc{Nibble}algorithm from Bhattacharya, Grandoni and Wajc [SODA'21] with the subsamplingtechnique of Kulkarni, Liu, Sah, Sawhney and Tarnawski [STOC'22].

Nibbling at Long Cycles: Dynamic (and Static) Edge Coloring in Optimal Time