Randomized greedy algorithms for independent sets and matchings in regular graphs: Exact results and finite girth corrections

We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant degree regular graphs. We show that for $r$-regular graphs with $n$ nodes and girth at least $g$, the algorithm finds an independent set of expected cardinality $f(r)n - O\big(\frac{(r-1)^{\frac{g}{2}}}{\frac{g}{2}!} n\big)$, where $f(r)$ is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings. As an implication we show that the greedy algorithm returns a nearly perfect matching when both the degree $r$ and girth $g$ are large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm in \emph{arbitrary} bounded degree graphs is concentrated around the mean. Finally, we analyze the performance of the greedy algorithm for the case of random i.i.d. weighted independent sets and matchings, and obtain a remarkably simple expression for the limiting expected values produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries from the results for the weighted case.

💡 Research Summary

The paper studies the performance of a very simple randomized greedy algorithm (called GREEDY) for constructing large independent sets and matchings in bounded‑degree regular graphs. The authors focus on r‑regular graphs whose girth (the length of the shortest cycle) is at least g, and they consider both the unweighted case and the case where vertices (for independent sets) or edges (for matchings) carry i.i.d. non‑negative continuous weights drawn from a distribution F with finite mean.

The main technical contribution is a precise analysis of GREEDY based on the “correlation decay” phenomenon. When the girth is large, the neighbourhood of any vertex up to distance ⌊(g‑2)/2⌋ is a tree, denoted T(r,r‑1,d). The authors introduce the notion of an influence‑blocking subgraph and prove that GREEDY’s decisions inside such a subgraph depend only on the subgraph itself, not on the rest of the graph. Consequently, the joint distribution of the algorithm’s choices for two vertices (or edges) that are far apart factorises up to an error term that decays as (r‑1)^{d}/d!. This reduction allows the problem to be transferred to an infinite r‑ary tree, where the greedy dynamics can be described by a simple recursion.

For the weighted case, the authors derive an explicit integral expression \