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Finding maximum matchings in random regular graphs in linear expected time
Author(s) -
Anastos Michael,
Frieze Alan
Publication year - 2021
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20980
Subject(s) - random graph , hopcroft–karp algorithm , matching (statistics) , combinatorics , time complexity , running time , mathematics , task (project management) , algorithm , binary logarithm , graph , computer science , discrete mathematics , line graph , pathwidth , statistics , management , economics
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 by Karp and Sipser suggest that the first algorithm is superior. In this paper we show that this is indeed the case, at least for random k ‐regular graphs. We show that w.h.p. the first algorithm will find a matching of size n / 2 − O ( log n ) in a random k ‐regular graph, k = O (1) . We also show that the algorithm can be adapted to find a maximum matching in O ( n ) time w.h.p. This is to be compared with O ( n 3/2 ) time for the worst‐case.