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#### A Faster Algorithm for Minimum Cycle Basis of Graphs

##### MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons45021

Mehlhorn,  Kurt
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45040

Michail,  Dimitrios
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

Telikepalli,  Kavitha
Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45155

Paluch,  Katarzyna
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

Díaz,  Josep
Max Planck Society;

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##### Zitation

Mehlhorn, K., Michail, D., Telikepalli, K., & Paluch, K. (2004). A Faster Algorithm for Minimum Cycle Basis of Graphs. In Automata, languages and programming: 31st International Colloquium, ICALP 2004 (pp. 846-857). Berlin, Germany: Springer.

In this paper we consider the problem of computing a minimum cycle basis in a graph $G$ with $m$ edges and $n$ vertices. The edges of $G$ have non-negative weights on them. The previous best result for this problem was an $O(m^{\omega}n)$ algorithm, where $\omega$ is the best exponent of matrix multiplication. It is presently known that $\omega < 2.376$. We obtain an $O(m^2n + mn^2\log n)$ algorithm for this problem. Our algorithm also uses fast matrix multiplication. When the edge weights are integers, we have an $O(m^2n)$ algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in $O(m^{\omega})$ time. For any $\epsilon > 0$, we also design a $1+\epsilon$ approximation algorithm to compute a cycle basis which is at most $1+\epsilon$ times the weight of a minimum cycle basis. The running time of this algorithm is $O(\frac{m^{\omega}}{\epsilon}\log(W/{\epsilon}))$ for reasonably dense graphs, where $W$ is the largest edge weight.