# Minimum Cycle Bases for Network Graphs

@article{Berger2004MinimumCB, title={Minimum Cycle Bases for Network Graphs}, author={Franziska Berger and Peter Gritzmann and Sven de Vries}, journal={Algorithmica}, year={2004}, volume={40}, pages={51-62} }

Abstract
The minimum cycle basis problem in a graph G = (V,E) is the task to construct a minimum length basis of its cycle vector space. A well-known algorithm by Horton of 1987 needs running time O(|V||E|2.376). We present a new combinatorial approach which generates minimum cycle bases in time O(\max{|E|3,|E||V|2log |V|}) with a space requirement of Θ(|E|2). This method is especially suitable for large sparse graphs of electric engineering applications since there, typically, |E| is close to…

## 61 Citations

New Approximation Algorithms for Minimum Cycle Bases of Graphs

- Computer ScienceAlgorithmica
- 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.

Minimum Cycle Basis, Algorithms & Applications

- Computer Science, Mathematics
- 2006

This work considers the problem of computing a minimum cycle basis of an undirected edge-weighted graph G with m edges and n vertices and presents an O(m2n + mn2 log n) algorithm to compute aminimum cycle basis and describes how to efficiently implement de Pina’s O( m3 +mn2 logn) algorithm.

Minimum path bases and relevant paths

- Business, EconomicsNetworks
- 2005

These findings provide support for the existence of a positive employment shadow after a transfer, whereas the shadow of succession hypothesis has to be rejected prior to transition.

Algorithms to Compute Minimum Cycle Basis in Directed Graphs

- Computer Science, MathematicsTheory 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.

A Polynomial Time Algorithm for Minimum Cycle Basis in Directed Graphs

- Mathematics, Computer ScienceSTACS
- 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.

Minimum Cycle Bases in Graphs Algorithms and Applications

- MathematicsMFCS
- 2007

A cycle basis of a graph is a family of cycles which spans all cycles of the graph, a set of edges with respect to which every vertex has even degree.

A Faster Deterministic Algorithm for Minimum Cycle Bases in Directed Graphs

- Computer Science, MathematicsICALP
- 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.

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

- Mathematics, Computer ScienceICALP
- 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.

Faster Algorithms for Minimum Cycle Basis in Directed Graphs

- Computer Science, MathematicsSIAM J. Comput.
- 2008

The fastest known algorithm for computing a minimum cycle basis in an undirected graph runs in $O(m^2n + mn^2\log n)$ time and the randomized algorithm for directed graphs matches this running time.

## References

SHOWING 1-10 OF 27 REFERENCES

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

- Mathematics, Computer ScienceSIAM 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.

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

- Mathematics, Computer ScienceSWAT
- 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.

The All-Pairs Min Cut Problem and the Minimum Cycle Basis Problem on Planar Graphs

- MathematicsSIAM J. Discret. Math.
- 1994

It is shown that on planar graphs, the all-pairs min cut (APMC) problem is equivalent to another problem, the minimum cycle basis (MCB) problem, on the dual graph, which leads to a new algorithm for solving both of these problems onPlanar graphs.

Minimum cycle bases of Halin graphs

- MathematicsJ. Graph Theory
- 2003

It is shown that all Halin graphs that are not “necklaces” have a unique minimum cycle basis.

An algorithm for finding a fundamental set of cycles of a graph

- MathematicsCACM
- 1969

A fast method is presented for finding a fundamental set of cycles for an undirected finite graph and is similar to that of Gotlieb and Corneil and superior to those of Welch and Welch.

Depth-First Search and Linear Graph Algorithms

- MathematicsSIAM J. Comput.
- 1972

The value of depth-first search or “backtracking” as a technique for solving problems is illustrated by two examples. An improved version of an algorithm for finding the strongly connected components…

Combinatorial relaxation algorithm for mixed polynomial matrices

- Mathematics, Computer ScienceMath. Program.
- 2001

An algorithm for computing the maximum degree of subdeterminants of a fixed order in a polynomial matrix that is obtained by substituting specific numerical values to the independent physical parameters of a mixed polynomial matrix is presented.