Local Algorithms for Bounded Degree Sparsifiers in Sparse Graphs.

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 eps u003e 0 be a slack parameter and alpha ge 1 be a density parameter.We devise local algorithms for computing:1. A (1+eps)-vertex cover sparsifier of degree O(alpha / eps), for any graph of arboricity alpha.footnote{In a graph of arboricity alpha the average degree of any induced subgraph is at most 2alpha.}2. A (1+eps)-maximum matching sparsifier and also a (1+eps)-maximal matching sparsifier of degree O(alpha / eps, for any graph of arboricity alpha.3. A (1+eps)-independent set sparsifier of degree O(alpha^2 / eps), for any graph of average degree alpha.Our algorithms require only a single communication round in the standard message passing model 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 / eps) or average degree O(d^2 / eps), at the expense of increasing the approximation guarantee by a factor of (1+eps). In particular, we can extend the plethora of recent local computation algorithms for approximate maximum and maximal matching from bounded degree graphs to bounded arboricity graphs with a negligible loss in the approximation guarantee.The inherently local behavior of our algorithms can be used to amplify the approximation guarantee of any sparsifier in time roughly linear in its size,which has immediate applications in the area of dynamic graph algorithms. In particular, the state-of-the-art algorithm formaintaining (2-eps)-vertex cover (VC) is at least linear in the graph size, even in dynamic forests. We provide a reduction from the dynamic to the static case, showing that if a t-VC can be computed from scratch in time T(n) in any (sub)family of graphs with arboricity bounded by alpha, for an arbitrary t ge 1,then a (t+eps)-VC can be maintained with update time frac{T(n)}{O((n / alpha) cdot eps^2)}, for any eps u003e 0. For planar graphs this yields an algorithm for maintaining a (1+eps)-VC with constant update time for any constant eps u003e 0.