Ariel Khuzman | Searcharxiv

arXiv2026

Efficient Parallel $(Δ+1)$-Edge-Coloring

We study the $(Δ+1)$-edge-coloring problem in the parallel $\left(\mathrm{PRAM}\right)$ model of computation. The celebrated Vizing's theorem [Viz64] states that every simple graph $G = (V,E)$ can be properly $(Δ+1)$-edge-colored. In a seminal paper, Karloff and Shmoys [KS87] devised a parallel algorithm with time $O\left(Δ^5\cdot\log n\cdot\left(\log^3 n+Δ^2\right)\right)$ and $O(m\cdotΔ)$ processors. This result was improved by Liang et al. [LSH96] to time $O\left(Δ^{4.5}\cdot \log^3Δ\cdot \log n + Δ^4 \cdot\log^4 n\right)$ and $O\left(n\cdotΔ^{3} +n^2\right)$ processors. [LSH96] claimed $O\left(Δ^{3.5} \cdot\log^3Δ\cdot \log n + Δ^3\cdot \log^4 n\right)$ time, but we point out a flaw in their analysis, which once corrected, results in the above bound. We devise a faster parallel algorithm for this fundamental problem. Specifically, our algorithm uses $O\left(Δ^4\cdot \log^4 n\right)$ time and $O(m\cdot Δ)$ processors. Another variant of our algorithm requires $O\left(Δ^{4+o(1)}\cdot\log^2 n\right)$ time, and $O\left(m\cdotΔ\cdot\log n\cdot\log^δΔ\right)$ processors, for an arbitrarily small $δ>0$. We also devise a few other tradeoffs between the time and the number of processors, and devise an improved algorithm for graphs with small arboricity. On the way to these results, we also provide a very fast parallel algorithm for updating $(Δ+1)$-edge-coloring. Our algorithm for this problem is dramatically faster and simpler than the previous state-of-the-art algorithm (due to [LSH96]) for this problem.

cs.DS↗

arXiv2024

Deterministic Simple $(Δ+\varepsilonα)$-Edge-Coloring in Near-Linear Time

We study the edge-coloring problem in simple $n$-vertex $m$-edge graphs with maximum degree $Δ$. This is one of the most classical and fundamental graph-algorithmic problems. Vizing's celebrated theorem provides $(Δ+1)$-edge-coloring in $O(m\cdot n)$ deterministic time. This running time was improved to $O\left(m\cdot\min\left\{Δ\cdot\log n,\sqrt{n}\right\}\right)$, and very recently to randomized $\tilde{O}\left(m\cdot n^{1/3}\right)$. A randomized $(1+\varepsilon)Δ$-edge-coloring algorithm can be computed in $O\left(m\cdot\frac{\log^6 n}{\varepsilon^2}\right)$ time, and for large values of $Δ$, this task requires randomized $O\left(\frac{m\cdot\log\varepsilon^{-1}}{\varepsilon^2}\right)$ time. It was however open if there exists a deterministic near-linear time algorithm for this basic problem. We devise a simple deterministic $(1+\varepsilon)Δ$-edge-coloring algorithm with running time $O\left(m\cdot\frac{\log n}{\varepsilon}\right)$. A randomized variant of our algorithm has running time $O(m\cdot(\varepsilon^{-18}+\log(\varepsilon\cdotΔ)))$. We also study edge-coloring of graphs with arboricity at most $α$. A randomized computation of $(Δ+1)$-edge-coloring requires $\tilde{O}\left(\min\{m\cdot\sqrt{n},m\cdotΔ\}\cdot\fracαΔ\right)$ time. Deterministically, this task can be done in $O\left(m\cdotα^7\cdot\log n\right)$ time. However, for large values of $α$, these algorithms require super-linear time. We devise a deterministic $(Δ+\varepsilonα)$-edge-coloring algorithm with running time $O\left(\frac{m\cdot\log n}{\varepsilon^7}\right)$. A randomized version of our algorithm requires $O\left(\frac{m\cdot\log n}{\varepsilon}\right)$ expected time. Our algorithm is based on a novel two-way degree-splitting, which we devise in this paper. We believe that this technique is of independent interest.

cs.DS↗