We prove new bounds on the distributed fractional coloring problem in the LOCAL model. Fractional $c$-colorings can be understood as multicolorings as follows. For some natural numbers $p$ and $q$ such that $p/q\leq c$, each node $v$ is assigned a set of at least $q$ colors from $\{1,\dots,p\}$ such that adjacent nodes are assigned disjoint sets of colors. The minimum $c$ for which a fractional $c$-coloring of a graph $G$ exists is called the fractional chromatic number $\chi_f(G)$ of $G$. Recently, [Bousquet, Esperet, and Pirot; SIROCCO '21] showed that for any constant $\epsilon>0$, a fractional $(\Delta+\epsilon)$-coloring can be computed in $\Delta^{O(\Delta)} + O(\Delta\cdot\log^* n)$ rounds. We show that such a coloring can be computed in only $O(\log^2 \Delta)$ rounds, without any dependency on $n$. We further show that in $O\big(\frac{\log n}{\epsilon}\big)$ rounds, it is possible to compute a fractional $(1+\epsilon)\chi_f(G)$-coloring, even if the fractional chromatic number $\chi_f(G)$ is not known. That is, this problem can be approximated arbitrarily well by an efficient algorithm in the LOCAL model. For the standard coloring problem, it is only known that an $O\big(\frac{\log n}{\log\log n}\big)$-approximation can be computed in polylogarithmic time in the LOCAL model. We also show that our distributed fractional coloring approximation algorithm is best possible. We show that in trees, which have fractional chromatic number $2$, computing a fractional $(2+\epsilon)$-coloring requires at least $\Omega\big(\frac{\log n}{\epsilon}\big)$ rounds. We finally study fractional colorings of regular grids. In [Bousquet, Esperet, and Pirot; SIROCCO '21], it is shown that in regular grids of bounded dimension, a fractional $(2+\epsilon)$-coloring can be computed in time $O(\log^* n)$. We show that such a coloring can even be computed in $O(1)$ rounds in the LOCAL model.
翻译:在 LOCAL 模式中, 我们证明在分布式正分色问题上有新的界限。 以美元计色的最小值, 以美元计色可以被理解为以下的多色。 对于某些自然数字, 美元和美元, 等于美元/ q\ leq c$, 每一个节点美元被分配到一套至少以美元为单位的颜色, $1,\ dots, 等于相邻节节点被分配不相交的颜色。 以美元计价的最小值$( 美元), 以美元计价的最小值 $G$; 对于某些自然数字, 美元/ 美元和美元, 每个节点美元 美元, 以美元计价的平价 。 以美元计价的方式, 以美元计价的平价數值來計價。 以美元表示, 任何數值的數值數值, 以美元為我們數的數值, 以美元數值為代數數, 以我們在數數數中, 以美元數數為代數的數數, 。