arXiv subjects

Nadav Panski

Publications and source records attributed to Nadav Panski.

3 recordsLinked to original sources

arXiv2023

Arboricity-Dependent Algorithms for Edge Coloring

The problem of edge coloring has been extensively studied over the years. Recently, this problem has received significant attention in the dynamic setting, where we are given a dynamic graph evolving via a sequence of edge insertions and deletions and our objective is to maintain an edge coloring of the graph. Currently, it is not known whether it is possible to maintain a $(\Delta+ O(\Delta^{1 - \mu}))$-edge coloring in $\tilde{O}(1)$ update time, for any constant $\mu > 0$, where $\Delta$ is the maximum degree of the graph. In this paper, we show how to efficiently maintain a $(\Delta + O(\alpha))$-edge coloring in $\tilde O(1)$ amortized update time, where $\alpha$ is the arboricty of the graph. Thus, we answer this question in the affirmative for graphs of sufficiently small arboricity.

cs.DS↗

arXiv2023

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

We consider the problem of maintaining a $(1+\epsilon)\Delta$-edge coloring in a dynamic graph $G$ with $n$ nodes and maximum degree at most $\Delta$. The state-of-the-art update time is $O_\epsilon(\text{polylog}(n))$, by Duan, He and Zhang [SODA'19] and by Christiansen [STOC'23], and more precisely $O(\log^7 n/\epsilon^2)$, where $\Delta = \Omega(\log^2 n / \epsilon^2)$. The following natural question arises: What is the best possible update time of an algorithm for this task? More specifically, \textbf{ can we bring it all the way down to some constant} (for constant $\epsilon$)? This question coincides with the \emph{static} time barrier for the problem: Even for $(2\Delta-1)$-coloring, there is only a naive $O(m \log \Delta)$-time algorithm. We answer this fundamental question in the affirmative, by presenting a dynamic $(1+\epsilon)\Delta$-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 first linear 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 subsampling technique of Kulkarni, Liu, Sah, Sawhney and Tarnawski [STOC'22].

cs.DS↗

arXiv2023

Density-Sensitive Algorithms for $(\Delta + 1)$-Edge Coloring

Vizing's theorem asserts the existence of a $(\Delta+1)$-edge coloring for any graph $G$, where $\Delta = \Delta(G)$ denotes the maximum degree of $G$. Several polynomial time $(\Delta+1)$-edge coloring algorithms are known, and the state-of-the-art running time (up to polylogarithmic factors) is $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot \Delta\})$, by Gabow et al.\ from 1985, where $n$ and $m$ denote the number of vertices and edges in the graph, respectively. (The $\tilde{O}$ notation suppresses polylogarithmic factors.) Recently, Sinnamon shaved off a polylogarithmic factor from the time bound of Gabow et al. The {arboricity} $\alpha = \alpha(G)$ of a graph $G$ is the minimum number of edge-disjoint forests into which its edge set can be partitioned, and it is a measure of the graph's "uniform density". While $\alpha \le \Delta$ in any graph, many natural and real-world graphs exhibit a significant separation between $\alpha$ and $\Delta$. In this work we design a $(\Delta+1)$-edge coloring algorithm with a running time of $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot \Delta\})\cdot \frac{\alpha}{\Delta}$, thus improving the longstanding time barrier by a factor of $\frac{\alpha}{\Delta}$. In particular, we achieve a near-linear runtime for bounded arboricity graphs (i.e., $\alpha = \tilde{O}(1)$) as well as when $\alpha = \tilde{O}(\frac{\Delta}{\sqrt{n}})$. Our algorithm builds on Sinnamon's algorithm, and can be viewed as a density-sensitive refinement of it.

cs.DS↗