Maximum matchings in sparse random graphs: Karp–Sipser revisited

Abstract

ABSTRACT:We study the average performance of a simple greedy algorithm for finding a matching in a sparse random graph G , where c ) 0 is constant. The algorithm was first n, c r n w proposed by Karp and Sipser Proceedings of the Twenty-Second Annual IEEE Symposium on x Foundations of Computing, 1981, pp. 364᎐375 . We give significantly improved estimates of the errors made by the algorithm. For the subcritical case where c -e we show that the algorithm finds a maximum matching with high probability. If c ) e then with high probability the algorithm produces a matching which is within n 1r5qoŽ1. of maximum size.

Maximum matchings in random bipartite graphs and the space utilization of Cuckoo Hash tables

“…We begin with a simple observation that is the basis of the Karp-Sipser Algorithm [16,2]. If v is a vertex of degree one in G and e is its unique incident edge, then there exists a maximum matching of G that includes e. Karp and Sipser exploited this via a simple greedy algorithm: Algorithm 1 Karp-Sipser Algorithm 1: procedure KSGreedy(G)…”

“…Then we randomly fill in the remaining dw − v 1 non-⋆ positions in x with values from some fixed v-subset R x of R, subject to each of these v vertices having degree at least two. The degrees Y i of these vertices will have the description described in the lemma (for a proof see Lemma 4 of [2]). …”

Finding a Maximum Matching in a Sparse Random Graph in O(n) Expected Time

“…, Z ν have the same distribution as the degrees of Γ 1 . This follows from Lemma 4 of [1]. If we choose λ so that E(P o(λ; ≥ 2) = 2µ ν or λ(e λ −1) e λ −1−λ = 2µ ν then the conditional probability, P(…”

Experimental Evaluation of the Greedy and Random Algorithms for Finding Independent Sets in Random Graphs

“…This difficulty already surfaces when one analyzes Greedy on random graphs with average degree 3. The only results that carry through the analysis of Greedy are those for finding matchings in random graphs ( [10], [1]) and for 3-regular random graphs ( [7]). However, it is not known how Greedy performs on random d-regular graphs with d ≥ 4; on random graphs with the average vertex degree d, for any d > 0; or on random graphs with a fixed edge-probability p > 0 (d = p(n − 1)).…”

“…The experiments described here involve running Random, Greedy, and Greedy m on randomly generated graphs from different domains: G(n, p) for a fixed p; G(n, p) for p = d/(n − 1), where d is an integer in [1,10]; and 3-regular graphs. All our experiments are reproducible, since we use a seeded pseudo-random number generator.…”