# An $\tilde{O}(m^{2}n)$ Algorithm for Minimum Cycle Basis of Graphs

@article{Kavitha2007An, title={An \$\tilde\{O\}(m^\{2\}n)\$ Algorithm for Minimum Cycle Basis of Graphs}, author={T. Kavitha and K. Mehlhorn and D. Michail and Katarzyna E. Paluch}, journal={Algorithmica}, year={2007}, volume={52}, pages={333-349} }

Abstract
We consider the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over
$\mathbb{F}_{2}$
generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle… Expand

#### 20 Citations

New Approximation Algorithms for Minimum Cycle Bases of Graphs

- Mathematics, Computer Science
- Algorithmica
- 2009

This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Θ(mω) bound, and two new algorithms to compute an approximate minimum cycle basis are presented. Expand

New Approximation Algorithms for Minimum Cycle Bases of Graphs

- Mathematics, Computer Science
- STACS
- 2007

This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Θ(mω) bound, and two new algorithms to compute an approximate minimum cycle basis are presented. Expand

Minimum cycle and homology bases of surface embedded graphs

- Mathematics, Computer Science
- SoCG
- 2016

The assumption of unique shortest paths can be avoided with high probability using randomization or deterministically by increasing the running time of the homology basis algorithm by a factor of $O(\log n)$. Expand

Minimum Cycle Basis and All-Pairs Min Cut of a Planar Graph in Subquadratic Time

- Mathematics, Computer Science
- ArXiv
- 2009

An output-sensitive algorithm is obtained that explicitly computes a minimum cycle basis in time and space for preprocessing the weight of a min cut between any two given vertices of $G$ and can be reported in constant time. Expand

On Finding a Minimum Weight Cycle Basis with Cycles of Bounded Length

- Mathematics, Computer Science
- CTW
- 2010

The Minimum Cycle Basis problem consists in finding a cycle basis C of minimum total weight w(C) = ∑ C∈C w (C), where the weight of a cycle is defined as w( c) =∑ e∈E the authors, and the problem has been extensively studied. Expand

Efficient Deterministic Algorithms for Finding a Minimum Cycle Basis in Undirected Graphs

- Mathematics, Computer Science
- IPCO
- 2010

This work revisits Horton's and de Pina's approaches and proposes a simple hybrid algorithm which improves the worst-case complexity to O(m2n / logn) and presents a very efficient related algorithm that relies on an adaptive independence test a la de PINA. Expand

Improved Minimum Cycle Bases Algorithms by Restriction to Isometric Cycles

- Mathematics
- 2011

We present improved algorithms for finding minimum cycle bases in undirected and directed graphs. For general graphs, the new algorithms are Monte Carlo and have running time O(mω), where m is the… Expand

Breaking the O(m2n) Barrier for Minimum Cycle Bases

- Mathematics, Computer Science
- ESA
- 2009

Improved algorithms for constructing minimum directed and undirected cycle bases in graphs are given with improved running times, based on the insight that the search for minimum bases can be restricted to a set of candidate cycles of total length O(nm). Expand

Minimum cycle bases of weighted outerplanar graphs

- Mathematics, Computer Science
- Inf. Process. Lett.
- 2010

This work gives the first optimal algorithm that computes a minimum cycle basis for any weighted outerplanar graph G to obtain an O(n)-space compact representation Z(C) for aminimum cycle basis C of G. Expand

Minimum Cycle Bases of Weighted Outerplanar Graphs

- Mathematics, Computer Science
- ISAAC
- 2009

This work gives the first known optimal algorithm that computes a minimum cycle basis for any weighted outerplanar graph, and gives an O(n)-time algorithm to obtain an O-space compact representation Z(?) for aminimum cycle basis of G. Expand

#### References

SHOWING 1-10 OF 25 REFERENCES

New Approximation Algorithms for Minimum Cycle Bases of Graphs

- Mathematics, Computer Science
- Algorithmica
- 2009

This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Θ(mω) bound, and two new algorithms to compute an approximate minimum cycle basis are presented. Expand

A Faster Algorithm for Minimum Cycle Basis of Graphs

- Computer Science, Mathematics
- ICALP
- 2004

A 1-e approximation algorithm to compute a cycle basis which is at most 1+e times the weight of a minimum cycle basis in a graph G with m edges and n vertices. Expand

A Polynomial-Time Algorithm to Find the Shortest Cycle Basis of a Graph

- Mathematics, Computer Science
- SIAM J. Comput.
- 1987

An algorithm is given that finds a cycle basis with the shortest possible length in $O(m^3 n)$ operations, which is the first known polynomial-time algorithm for this problem. Expand

When Do Short Cycles Generate the Cycle Space?

- Computer Science, Mathematics
- J. Comb. Theory, Ser. B
- 1993

Those graphs for which {C(e) | e ∈ E} is a cycle basis (hence, the cycle basis of minimum weight) for every perturbed edge weighting are characterized. Expand

Minimum Path Bases

- Mathematics, Computer Science
- J. Algorithms
- 1993

A new vector space associated with the paths and cycles in a graph is introduced and a polynomial algorithm for finding a minimum weight basis for this space is presented, which can be implemented with worst case time complexity O. Expand

Algorithms for Generating Fundamental Cycles in a Graph

- Mathematics, Computer Science
- TOMS
- 1982

It is shown that for regular graphs of order n the expected value of the total length of a minimum fundamentalcycle set does not exceed O(n2). Expand

All-pairs small-stretch paths

- Mathematics, Computer Science
- SODA '97
- 1997

Three algorithms for finding small-stretch paths between all pairs of vertices in a weighted graph with n vertices and m edges are described. Expand

A Polynomial Time Algorithm to Find the Minimum Cycle Basis of a Regular Matroid

- Mathematics, Computer Science
- SWAT
- 2002

An algorithm is given to solve the minimum cycle basis problem for regular matroids based upon Seymour's decomposition theorem, the Gomory-Hu tree, which is essentially the solution for cographicMatroids; and the corresponding result for graphs. Expand

All Pairs Shortest Paths for Graphs with Small Integer Length Edges

- Mathematics, Computer Science
- J. Comput. Syst. Sci.
- 1997

This paper shows how to transform these algorithms to solve the all pairs shortest paths (APSP), in the same time complexity, up to a polylogarithmic factor. Expand

On optimally sparse cycle and coboundary basis for a linear graph

- Computer Science
- 1973

A graph-theoretic study of the computational efficiency of the generalized loop analysis and the generalized cutset analysis is presented. It is shown that the choice of an optimum mode of analysis… Expand