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 GΔ be a random subgraph of G that includes, for each vertex v ∈ G, Δ random edges incident on it. We show that, with high probability, GΔ is a (1+ε)-matching sparsifier for G, i.e., the maximum matching size of GΔ is within a factor of 1+ε from that of G. One can then work on the sparsifier GΔ rather than on the original graph G. Since GΔ 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]. GΔ 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 √n/β, for constant ε) update time of O(√β n).