Randomized greedy algorithm for independent sets in regular uniform hypergraphs with large girth

Abstract

In this paper, we consider a randomized greedy algorithm for independent sets in r-uniform d-regular hypergraphs G on n vertices with girth g. By analyzing the expected size of the independent sets generated by this algorithm, we show that (G) ≥ (f ( , r) − (g, , r))n, where (g, , r) converges to 0 as g → ∞ for fixed d and r, and f (d, r) is determined by a differential equation. This extends earlier results of Garmarnik and Goldberg for graphs [8]. We also prove that when applying this algorithm to uniform linear hypergraphs with bounded degree, the size of the independent sets generated by this algorithm concentrate around the mean asymptotically almost surely.

The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs

“…Here we relate our result to that of Nie and Verstraëte [22]. Let H be a (deterministic) ∆-regular r-uniform hypergraph with n vertices.…”

“…Krivelevich, Mészáros, Michaeli and Shikhelman [18] gave a very general analysis of the independent process, showing in many cases (sequences of graphs) of interest that the final size of the independent set can be approximated using an appropriate "limiting" object (locally finite graph). While there is no formal statement or proof of a hypergraph analog of the result in [18], the authors do show that, assuming such an analog is true, it implies something very similar to Nie and Verstraëte's result in [22]. Indeed, as noted in [18], the local limit (see [18] for technical definitions) of a sequence of r-uniform ∆-regular hypergraphs with girth tending to infinity is T r ∆ , the infinite rooted r-uniform ∆-regular loose tree.…”

“…Gamarnik and Goldberg [14] then established concentration of the final size of the independent set or matching produced by the processes (again working on regular graphs of high girth). Nie and Verstraëte [22] also analyzed the independent process on regular hypergraphs of large girth. Since sparse random regular hypergraphs have very few short cycles, it is natural to expect that the final independent set obtained in the setting of [22] should be asymptotically the same as ours.…”

“…Nie and Verstraëte [22] also analyzed the independent process on regular hypergraphs of large girth. Since sparse random regular hypergraphs have very few short cycles, it is natural to expect that the final independent set obtained in the setting of [22] should be asymptotically the same as ours. Indeed, in subsection 1.3.2 we will see that this is the case as well.…”

“…Informally this means that if we choose a vertex v in an r-uniform ∆-regular hypergraph of girth say 3k, the ball of radius k around v is isomorphic to the ball of radius k around a vertex of T r ∆ . Using the limiting object T r ∆ to set up a differential equation recovers the result of [22] (ignoring the precise error bound in the result).…”