Fast algorithms for Vizing's theorem on bounded degree graphs

Vizing's theorem states that every graph $G$ of maximum degree $\Delta$ can be properly edge-colored using $\Delta + 1$ colors. The fastest currently known $(\Delta+1)$-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time $O(m\sqrt{n})$, where $n :=|V(G)|$ and $m :=|E(G)|$. In this paper we investigate the case when $\Delta$ is constant, i.e., $\Delta = O(1)$. In this regime, the running time of Sinnamon's algorithm is $O(n^{3/2})$, which can be improved to $O(n \log n)$, as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only $O(n)$, which is obviously best possible. We also develop new algorithms for $(\Delta+1)$-edge-coloring in the $\mathsf{LOCAL}$ model of distributed computation. Namely, we design a deterministic $\mathsf{LOCAL}$ algorithm with running time $\tilde{O}(\log^5 n)$ and a randomized $\mathsf{LOCAL}$ algorithm with running time $O(\log ^2 n)$. All these results are new already for $\Delta = 4$. Although our focus is on the constant $\Delta$ regime, our results remain interesting for $\Delta$ up to $\log^{o(1)} n$. The key new ingredient in our algorithms is a novel application of the entropy compression method.Comment: 42 pages, 15 figure

Faster Deterministic Distributed MIS and Approximate Matching

$\renewcommand{\tilde}{\widetilde}$We present an $\tilde{O}(\log^2 n)$ round deterministic distributed algorithm for the maximal independent set problem. By known reductions, this round complexity extends also to maximal matching, $\Delta+1$ vertex coloring, and $2\Delta-1$ edge coloring. These four problems are among the most central problems in distributed graph algorithms and have been studied extensively for the past four decades. This improved round complexity comes closer to the $\tilde{\Omega}(\log n)$ lower bound of maximal independent set and maximal matching [Balliu et al. FOCS '19]. The previous best known deterministic complexity for all of these problems was $\Theta(\log^3 n)$. Via the shattering technique, the improvement permeates also to the corresponding randomized complexities, e.g., the new randomized complexity of $\Delta+1$ vertex coloring is now $\tilde{O}(\log^2\log n)$ rounds. Our approach is a novel combination of the previously known two methods for developing deterministic algorithms for these problems, namely global derandomization via network decomposition (see e.g., [Rozhon, Ghaffari STOC'20; Ghaffari, Grunau, Rozhon SODA'21; Ghaffari et al. SODA'23]) and local rounding of fractional solutions (see e.g., [Fischer DISC'17; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Ghaffari, Kuhn FOCS'21; Faour et al. SODA'23]). We consider a relaxation of the classic network decomposition concept, where instead of requiring the clusters in the same block to be non-adjacent, we allow each node to have a small number of neighboring clusters. We also show a deterministic algorithm that computes this relaxed decomposition faster than standard decompositions. We then use this relaxed decomposition to significantly improve the integrality of certain fractional solutions, before handing them to the local rounding procedure that now has to do fewer rounding steps

Streaming and Massively Parallel Algorithms for Edge Coloring

A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. (Note that the maximum degree, Delta, is a trivial lower bound.) In this paper, we revisit this fundamental problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model: - Massively Parallel Computation: We give a randomized MPC algorithm that with high probability returns a Delta+O~(Delta^(3/4)) edge coloring in O(1) rounds using O(n) space per machine and O(m) total space. The space per machine can also be further improved to n^(1-Omega(1)) if Delta = n^Omega(1). Our algorithm improves upon a previous result of Harvey et al. [SPAA 2018]. - Graph Streaming: Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors. We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors with high probability if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors

On the Complexity of Distributed Splitting Problems

One of the fundamental open problems in the area of distributed graph algorithms is the question of whether randomization is needed for efficient symmetry breaking. While there are fast, $\text{poly}\log n$-time randomized distributed algorithms for all of the classic symmetry breaking problems, for many of them, the best deterministic algorithms are almost exponentially slower. The following basic local splitting problem, which is known as the \emph{weak splitting} problem takes a central role in this context: Each node of a graph $G=(V,E)$ has to be colored red or blue such that each node of sufficiently large degree has at least one node of each color among its neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly simple problem is complete w.r.t. the above fundamental open question in the following sense: If there is an efficient $\text{poly}\log n$-time determinstic distributed algorithm for weak splitting, then there is such an algorithm for all locally checkable graph problems for which an efficient randomized algorithm exists. In this paper, we investigate the distributed complexity of weak splitting and some closely related problems. E.g., we obtain efficient algorithms for special cases of weak splitting, where the graph is nearly regular. In particular, we show that if $\delta$ and $\Delta$ are the minimum and maximum degrees of $G$ and if $\delta=\Omega(\log n)$, weak splitting can be solved deterministically in time $O\big(\frac{\Delta}{\delta}\cdot\text{poly}(\log n)\big)$. Further, if $\delta = \Omega(\log\log n)$ and $\Delta\leq 2^{\varepsilon\delta}$, there is a randomized algorithm with time complexity $O\big(\frac{\Delta}{\delta}\cdot\text{poly}(\log\log n)\big)$