A Unified Sparsification Approach for Matching Problems in Graphs of Bounded Neighborhood Independence
Lazar Milenković, Shay Solomon · 2020
The neighborhood independence number of a graph G, denoted by β = β(G), is the size of the largest independent set in the neighborhood of any vertex. Graphs with bounded neighborhood independence, already for constant β, constitute a wide family of possibly dense graphs, including line graphs, unit-disk graphs, claw-free graphs and graphs of bounded growth, which has been well-studied in the area of distributed computing. In ICALP'19, Assadi and Solomon [8] showed that, for any n-vertex graph G, a maximal matching can be computed in O(n log n · β) time in the classic sequential setting. This result shows that, surprisingly, for almost the entire regime of parameter β, a maximal matching can be computed much faster than reading the entire input. The algorithm of [8], however, is inherently sequential and centralized. Moreover, a maximal matching provides a 2-approximate (maximum) matching, and the question of whether a better-than-2-approximate matching can be computed in sublinear time remained open.