Implementing Minimum Cycle Basis Algorithms

In this paper we consider the problem of computing a minimum cycle basis of an undirected graph $G = (V,E)$ with $n$ vertices and $m$ edges. We describe an efficient implementation of an $O(m^3 + mn^2\log n)$ algorithm presented in~\cite{PINA95}. For sparse graphs this is the currently best known algorithm. This algorithm's running time can be partitioned into two parts with time $O(m^3)$ and $O( m^2n + mn^2 \log n)$ respectively. Our experimental findings imply that the true bottleneck of a sophisticated implementation is the $O( m^2 n + mn^2 \log n)$ part. A straightforward implementation would require $\Omega(nm)$ shortest path computations, thus we develop several heuristics in order to get a practical algorithm. Our experiments show that in random graphs our techniques result in a significant speedup. Based on our experimental observations, we combine the two fundamentally different approaches to compute a minimum cycle basis used in~\cite{PINA95,KMMP04} and~\cite{HOR87,MATR02}, to obtain a new hybrid algorithm with running time $O( m^2 n^2 )$. The hybrid algorithm is very efficient in practice for random dense unweighted graphs. Finally, we compare these two algorithms with a number of previous implementations for finding a minimum cycle basis in an undirected graph