Efficient approximation algorithms for shortest cycles in undirected graphs - Lingas, Andrzej; Lundell, Eva-Marta (Information processing letters, 2009)
We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. Given an adjacency-list representation of an undirected graph G with n vertices and unknown girth k, our algorithm returns with high probability a cycle of length at most 2 k for even k and 2 k + 2 for odd k, in time O ( n 3 2 log n ) . Thus, in general, it yields a 2 2 3 approximation. For a weighted, undirected graph, with non-negative edge weights in the range { 1 , 2 , … , M } , we present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle that runs in time O ( n 2 log n ( log n + log M ) ) .