A polynomial-time algorithm to find the shortest cycle basis of a graph
SIAM Journal on Computing
A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem
SIAM Journal on Discrete Mathematics
A Polynomial Time Algorithm to Find the Minimum Cycle Basis of a Regular Matroid
SWAT '02 Proceedings of the 8th Scandinavian Workshop on Algorithm Theory
Minimum Cycle Bases for Network Graphs
Algorithmica
Graph Theory with Applications to Engineering and Computer Science (Prentice Hall Series in Automatic Computation)
Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Implementing minimum cycle basis algorithms
Journal of Experimental Algorithmics (JEA)
Meshing genus-1 point clouds using discrete one-forms
Computers and Graphics
Faster Algorithms for Minimum Cycle Basis in Directed Graphs
SIAM Journal on Computing
A greedy approach to compute a minimum cycle basis of a directed graph
Information Processing Letters
New approximation algorithms for minimum cycle bases of graphs
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
An improved heuristic for computing short integral cycle bases
Journal of Experimental Algorithmics (JEA)
Minimum Cycle Bases and Their Applications
Algorithmics of Large and Complex Networks
Minimum Cycle Bases of Weighted Outerplanar Graphs
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Finding good cycle constraints for large scale multi-robot SLAM
ICRA'09 Proceedings of the 2009 IEEE international conference on Robotics and Automation
Minimum cycle bases of weighted outerplanar graphs
Information Processing Letters
Properties of Gomory-Hu co-cycle bases
Theoretical Computer Science
Efficient deterministic algorithms for finding a minimum cycle basis in undirected graphs
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Survey: Cycle bases in graphs characterization, algorithms, complexity, and applications
Computer Science Review
Integral cycle bases for cyclic timetabling
Discrete Optimization
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We consider the problem of computing exact or approximate minimum cycle bases of an undirected (or directed) graph G with m edges, n vertices and nonnegative edge weights. In this problem, a {0, 1} (−1,0,1}) incidence vector is associated with each cycle and the vector space over F2 (Q) 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 basis of G. Cycle bases of low weight are useful in a number of contexts, for example, the analysis of electrical networks, structural engineering, chemistry, and surface reconstruction. There exists a set of Θ(mn) cycles which is guaranteed to contain a minimum cycle basis. A minimum basis can be extracted by Gaussian elimination. The resulting algorithm [Horton 1987] was the first polynomial-time algorithm. Faster and more complicated algorithms have been found since then. We present a very simple method for extracting a minimum cycle basis from the candidate set with running time O(m2 n), which improves the running time for sparse graphs. Furthermore, in the undirected case by using bit-packing we improve the running time also in the case of dense graphs. For undirected graphs we derive an O(m2 n/log n + n2 m) algorithm. For directed graphs we get an O(m3 n) deterministic and an O(m2 n) randomized algorithm. Our results improve the running times of both exact and approximate algorithms. Finally, we derive a smaller candidate set with size in Ω(m) ∩ O(mn).