On the cycle polytope of a binary matroid
Journal of Combinatorial Theory Series B
A polynomial-time algorithm to find the shortest cycle basis of a graph
SIAM Journal on Computing
Very simple methods for all pairs network flow analysis
SIAM Journal on Computing
Cycle bases from orderings and coverings
Discrete Mathematics
A Graph-Theoretic Game and its Application to the $k$-Server Problem
SIAM Journal on Computing
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
The zoo of tree spanner problems
Discrete Applied Mathematics
ISAAC '01 Proceedings of the 12th International Symposium on Algorithms and Computation
On the minimum diameter spanning tree problem
Information Processing Letters
Benchmarks for strictly fundamental cycle bases
WEA'07 Proceedings of the 6th international conference on Experimental algorithms
Cyclic matrices of weighted digraphs
Discrete Applied Mathematics
Properties of Gomory-Hu co-cycle bases
Theoretical Computer Science
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Given an undirected, connected network G=(V,E) with weights on the edges, the cut basis problem is asking for a maximal number of linear independent cuts such that the sum of the cut weights is minimized. Surprisingly, this problem has not attained as much attention as another graph theoretic problem closely related to it, namely, the cycle basis problem. We consider two versions of the problem: the unconstrained and the fundamental cut basis problem. For the unconstrained case, where the cuts in the basis can be of an arbitrary kind, the problem can be written as a multiterminal network flow problem, and is thus solvable in strongly polynomial time. In contrast, the fundamental cut basis problem, where all cuts in the basis are obtained by deleting an edge, each from a spanning tree T, is shown to be NP-hard. In this proof, we also show that a tree which induces the minimum fundamental cycle basis is also an optimal solution for the minimum fundamental cut basis problem in unweighted graphs. We present heuristics, integer programming formulations and summarize first experiences with numerical tests.