# A Faster Algorithm for Minimum Cycle Basis of Graphs

@inproceedings{Kavitha2004AFA, title={A Faster Algorithm for Minimum Cycle Basis of Graphs}, author={T. Kavitha and K. Mehlhorn and D. Michail and Katarzyna E. Paluch}, booktitle={ICALP}, year={2004} }

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 ω n) algorithm, where ω is the best exponent of matrix multiplication. It is presently known that ω 0, we also design a 1+e approximation algorithm to compute a cycle basis which is at most 1+e times the weight of a minimum cycle basis. The running time of this algorithm is \(O… Expand

#### 64 Citations

An Õ(mn) Algorithm for Minimum Cycle Basis of Graphs∗

- 2004

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… Expand

An Õ(m2n) Randomized Algorithm to Compute a Minimum Cycle Basis of a Directed Graph

- Mathematics, Computer Science
- ICALP
- 2005

A simple O(m2n) randomized algorithm for the problem of computing a minimum cycle basis in an undirected graph and this algorithm almost matches the fastest known algorithm for this problem. 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

Algorithms to Compute Minimum Cycle Basis in Directed Graphs

- Mathematics, Computer Science
- Theory of Computing Systems
- 2006

An $\tilde{O}(m^4n)$ algorithm is presented, which is the first polynomial-time algorithm for computing a minimum cycle basis in G. Expand

A Polynomial Time Algorithm for Minimum Cycle Basis in Directed Graphs

- Mathematics, Computer Science
- STACS
- 2005

An O(m4n) algorithm is given, which is the first polynomial time algorithm for this problem of computing a minimum cycle basis in a directed graph G with m arcs and n vertices. Expand

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

- Computer Science, Mathematics
- Algorithmica
- 2007

This work considers the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices and designs an O(m2n+mn2log n) algorithm and an 1+ε approximation algorithm. Expand

A Faster Deterministic Algorithm for Minimum Cycle Bases in Directed Graphs

- Mathematics, Computer Science
- ICALP
- 2006

This work considers the problem of computing a minimum cycle basis in a directed graph G whose edges have non-negative weights and presents an O(m3n + m2n2logn) algorithm, a slightly improved running time improvement of the current fastest randomized algorithm. Expand

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

Minimum Cycle Basis, Algorithms & Applications

- 2006

We consider the problem of computing a minimum cycle basis of an undirected edge-weighted graph G with m edges and n vertices. In this problem, a {0, 1} incidence vector is associated with each cycle… Expand

Faster Randomized and Deterministic Algorithms for Minimum Cycle Bases in Directed Graphs ∗

- 2006

We consider the problem of computing a minimum cycle basis in a directed graph. The input to this problem is a directed graph G whose edges have non-negative weights. A cycle in this graph is… Expand

#### References

SHOWING 1-10 OF 18 REFERENCES

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

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

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

On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs

- Computer Science, Mathematics
- J. Comput. Syst. Sci.
- 1995

We present an algorithm, APD, that solves the distance version of the all-pairs-shortest-path problem for undirected, unweighted n-vertex graphs in time O(M(n) log n), where M(n) denotes the time… Expand

Undirected single-source shortest paths with positive integer weights in linear time

- Mathematics, Computer Science
- JACM
- 1999

A deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with positive integer weights, which avoids the sorting bottleneck by building a hierarchical bucketing structure. Expand

Odd Minimum Cut-Sets and b-Matchings

- Mathematics, Computer Science
- Math. Oper. Res.
- 1982

We show that the determination of a minimum cut-set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of Gomory… 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

All pairs shortest paths in weighted directed graphs-exact and almost exact algorithms

- Mathematics, Computer Science
- Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

Two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs using fast matrix multiplication algorithms are presented. Expand

Cycle bases of minimal measure for the structural analysis of skeletal structures by the flexibility method

- Mathematics
- Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1976

The development of the flexibility method of analysis of skeletal structures has been hindered by the difficulty of determining a suitable statical basis on which to form the flexibility matrix. A… Expand