A Unified Sparsification Approach for Matching Problems in Graphs of Bounded Neighborhood Independence

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. In this work we propose a unified and surprisingly simple approach for producing (1+ϵ)-approximate matchings, for arbitrarily small ϵ >0. Specifically, set Δ= O(β/ϵ log 1/ϵ) and let G be a random subgraph of G that includes, for each vertex v ĝ G, Δrandom edges incident on it. We show that, with high probability, G is a (1+ϵ)-matching sparsifier for G, i.e., the maximum matching size of G is within a factor of 1+ϵ from that of G. One can then work on the sparsifier G rather than on the original graph G. Since can be implemented efficiently in various settings, this approach is of broad applicability; some concrete implications are: A (1+ϵ)-approximate matching can be computed in the classic sequential setting in O(n/ · β ϵ² · log 1/ϵ) time, shaving a log n factor from the runtime of [8] (for any constant ϵ), and more importantly achieving an approximation factor of 1+ϵ rather than 2. For constant ϵ, our runtime is tight, matching a lower bound of ω(n · β) due to [5,8]. G can be computed in a single communication round in distributed networks. Consequently, a (1+ϵ)-approximate matching can be computed in (β/ϵ log 1/ϵ)^O (1/ϵ),+, O(1/ ϵ ²),·, log∗n [[L: We changed β/ϵ² to β/ϵ here.]] communications rounds, which reduces to O(log∗n) rounds when β and ϵ are constants; the previous (deterministic) algorithm by Barenboim and Oren [16,17] requires a similar number of rounds but its approximation factor is 2+ϵ. Our sparsifier also provides a rare example of an algorithm achieving a sublinear message complexity. A (1+ϵ)-approximate matching can be dynamically maintained with update time O(β/ϵ³ log 1/ϵ); the previous (deterministic) algorithm by Barenboim and Maimon [14] achieves approximation factor 2 with a higher (by a factor of β, for constant ϵ) update time of O(β n).