Local Algorithms for Bounded Degree Sparsifiers in Sparse Graphs

Abstract: 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 [...]