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Abstract:
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.
