Karp–Sipser on Random Graphs with a Fixed Degree Sequence

Tom Bohman,Alan Frieze

doi:10.1017/s0963548311000265

Abstract

Let Δ ≥ 3 be an integer. Given a fixed z ∈ +Δ such that zΔ > 0, we consider a graph Gz drawn uniformly at random from the collection of graphs with zin vertices of degree i for i = 1,. . .,Δ. We study the performance of the Karp–Sipser algorithm when applied to Gz. If there is an index δ > 1 such that z1 = . . . = zδ−1 = 0 and δzδ,. . .,ΔzΔ is a log-concave sequence of positive reals, then with high probability the Karp–Sipser algorithm succeeds in finding a matching with n ∥ z ∥ 1/2 − o(n1−ε) edges in Gz, where ε = ε (Δ, z) is a constant.

Highlights

The Karp-Sipser algorithm is the following randomized algorithm for finding a large matching in a graph
The point where the solution to the system of the differential equations (2) leaves the interior of D is either at the origin or at a point z with φ(z) = 1. Note that these two outcomes correspond to success and failure, respectively, of the Karp-Sipser algorithm applied to the graph Y(0): If we arrive at a distribution in the vicinity of the origin most vertices are saturated by the matching produced by the Karp-Sipser algorithm and if we arrive at a distribution with φ > 1 pendant edges quickly accumulate and Karp-Sipser does not produce an almost perfect matching
Theorem 1 can be extended to apply to degree distributions z with z1 > 0

Summary

Introduction

The Karp-Sipser algorithm is the following randomized algorithm for finding a large matching in a graph. (N.b. there are other degree distributions for which we can see that the Karp-Sipser algorithm does not produce an almost perfect matching. If ∆ ≥ 3 and z ∈ R∆+ such that z1 = 0, z∆ > 0 and z satisfies (1) whp the Karp-Sipser algorithm produces a matching with (1 − o (n− )) z 1n/2 edges in Gz for some constant = (∆, z). (This issue is discussed in more detail in Section 2 below.) the main contribution of Theorem 1 is an analytic proof that the solution of the associated system of differential equations is well-behaved all the way to the termination of the algorithm for every distribution z that satisfies the given conditions. The Conclusion contains an extension of Theorem 1 to random graphs Gz with z1 > 0, a discussion of the matching number of the random graph Gz and a number of open questions

Proof of Theorem 1

Proof of Theorem 2

Proof of Lemma 5

Conclusion