In graph sparsification, the goal has almost always been of {global} nature: compress a graph into a smaller subgraph ({sparsifier}) that maintains certain features of the original graph. Algorithms can then run on the sparsifier, which in many cases leads to improvements in the overall runtime and memory. This paper studies sparsifiers that have bounded (maximum) degree, and are thus {locally} sparse, aiming to improve local measures of runtime and memory. To improve those local measures, it is important to be able to compute such sparsifiers {locally}. We initiate the study of local algorithms for bounded degree sparsifiers in unweighted sparse graphs, focusing on the problems of vertex cover, matching, and independent set. Let $\epsilon > 0$ be a slack parameter and $\alpha \ge 1$ be a density parameter. We devise local algorithms for computing: (1) A $(1+\epsilon)$-vertex cover sparsifier of degree $O(\alpha / \epsilon)$, for any graph of {arboricity} $\alpha$. (2) A $(1+\epsilon)$-maximum matching sparsifier and also a $(1+\epsilon)$-maximal matching sparsifier of degree $O(\alpha / \epsilon)$, for any graph of arboricity $\alpha$. (3) A $(1+\epsilon)$-independent set sparsifier of degree $O(\alpha^2 / \epsilon)$, for any graph of average degree $\alpha$. Our algorithms require only a single communication round in the standard message passing models of distributed computing, and moreover, they can be simulated locally in a trivial way. As an immediate application we can extend results from distributed computing and local computation algorithms that apply to graphs of degree bounded by $d$ to graphs of arboricity $O(d / \epsilon)$ or average degree $O(d^2 / \epsilon)$, at the expense of increasing the approximation guarantee by a factor of $(1+\epsilon)$. In particular, we can extend the plethora of recent local computation algorithms [...]
翻译:在图形垃圾化中, 目标几乎总是 {global} 性质 : 将一个图形压缩成一个小的子集({sparsiter}}) 以保持原始图形的某些特性。 ALgorithms 可以在总运行时间和记忆量上运行, 在许多情况下, 这会导致整个运行时间和记忆度的改善。 这个纸学修饰器已经绑定( 最大) 度, 因此是 {ge} 稀少, 目的是改进运行时间和记忆的本地度度测量。 为了改进这些本地度测量, 我们必须能够将一个小的 $( comlial2) 压缩成一个小的子集成( $ 美元 ) 。 我们开始研究本地度的 美元 1 美元 的平面值, 平面 1 平面 平面 平面的 美元 。