Finding perfect matchings in random cubic graphs in linear time

TL;DR

This paper proves that a specific algorithm efficiently finds near-perfect and perfect matchings in random cubic graphs with high probability, significantly improving the runtime from previous worst-case bounds.

Contribution

It demonstrates that the first Karp-Sipser algorithm is highly effective for random cubic graphs and can be adapted to find perfect matchings in linear time.

Findings

The algorithm finds a matching of size n/2 - O(log n) w.h.p.

It can be adapted to find perfect matchings in O(n) time.

Empirical results support the algorithm's superiority in random cubic graphs.

Abstract

In a seminal paper on finding large matchings in sparse random graphs, Karp and Sipser proposed two algorithms for this task. The second algorithm has been intensely studied, but due to technical difficulties, the first algorithm has received less attention. Empirical results in \cite{KS} suggest that the first algorithm is superior. In this paper we show that this is indeed the case, at least for random cubic graphs. We show that w.h.p. the first algorithm will find a matching of size $n/2 - O(\log n)$ on a random cubic graph (indeed on a random graph with degrees in $\{3,4\}$). We also show that the algorithm can be adapted to find a perfect matching w.h.p. in $O(n)$ time, as opposed to $O(n^{3/2})$ time for the worst-case.