A polynomial-time algorithm to find the shortest cycle basis of a graph
SIAM Journal on Computing
A data structure for dynamic trees
Journal of Computer and System Sciences
Very simple methods for all pairs network flow analysis
SIAM Journal on Computing
The All-Pairs Min Cut Problem and the Minimum Cycle Basis Problem on Planar Graphs
SIAM Journal on Discrete Mathematics
Beyond the flow decomposition barrier
Journal of the ACM (JACM)
Algorithms for Generating Fundamental Cycles in a Graph
ACM Transactions on Mathematical Software (TOMS)
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
Meshing genus-1 point clouds using discrete one-forms
Computers and Graphics
Discrete Applied Mathematics
Algorithms to Compute Minimum Cycle Basis in Directed Graphs
Theory of Computing Systems
Minimum cycle bases: Faster and simpler
ACM Transactions on Algorithms (TALG)
A greedy approach to compute a minimum cycle basis of a directed graph
Information Processing Letters
Minimum cut bases in undirected networks
Discrete Applied Mathematics
An Õ(m2n) randomized algorithm to compute a minimum cycle basis of a directed graph
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
A faster deterministic algorithm for minimum cycle bases in directed graphs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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
Hi-index | 5.23 |
Let D=(V,A) be a directed graph with non-negative arc weights. We study the problem of computing certain special co-cycle bases of D, in particular, a minimum weight weakly fundamental co-cycle basis. A co-cycle in D corresponds to a cut in the underlying undirected graph and a {-1,0,1} arc incidence vector is associated with each co-cycle, where the +/-1 coordinates are used for the arcs crossing the cut. The weight of a co-cycle C is the sum of the weights of those arcs a such that C(a)=+/-1. The vector space over Q generated by the arc incidence vectors of the co-cycles is the co-cycle space of D. The co-cycle space of D can also be defined as the orthogonal complement of the cycle space of D. A set of linearly independent co-cycles that span the co-cycle space is a co-cycle basis of D. The problem of computing a co-cycle basis {C"1,...,C"k} such that the sum of weights of the co-cycles in the basis is the least possible is the minimum co-cycle basis problem. A co-cycle basis {C"1,...,C"k} is weakly fundamental if for every i there is an arc a"i such that C"i(a"i)=+/-1 while C"j(a"i)=0 for ji. The minimum cycle basis problem in directed and undirected graphs is a well-studied problem and while polynomial time algorithms are known for these problems, the problem of computing a minimum weight weakly fundamental cycle basis has recently been shown to be APX-hard. We show that the co-cycle basis corresponding to the cuts of a Gomory-Hu tree T of the underlying undirected graph of D is a minimum weight weakly fundamental co-cycle basis of D. This is, in fact, a minimum co-cycle basis of D and it is also totally unimodular. Thus this is a special co-cycle basis that simultaneously answers several questions in the domain of co-cycle bases. It is known that there is no such special cycle basis for general graphs.