A construction for binary matroids
Discrete Mathematics
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
A faster algorithm for finding the minimum cut in a graph
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Counterexamples for Directed and Node Capacitated Cut-Trees
SIAM Journal on Computing
The primal-dual method for approximation algorithms and its application to network design problems
Approximation algorithms for NP-hard problems
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We give algorithms for the directed minimum odd or even cut problem and certain generalizations. Our algorithms improve on the previous best ones of Goemans and Ramakrishnan by a factor of O(n) (here n is the size of the ground vertex set). Our improvements apply among others to the minimum directed Todd or T-even cut and to the directed minimum Steiner cut problems. The (slightly more general) result of Goemans and Ramakrishnan shows that a collection of minimal minimizers of a submodular function (i.e. minimum cuts) contains the odd minimizers. In contrast our algorithm selects an n-times smaller class of not necessarily minimal minimizers and out of these sets we construct the odd minimizer. If M(n,m) denotes the time of a u-v minimum cut computation in a directed graph with n vertices and m edges, then we may find a directed minimum - odd or T-odd cut with V (or T) even in O(n2m + n ċ M(n,m)) time; - even or T-even cut in O(n3m + n2 ċ M(n, m)) time. The key of our construction is a so-called parity uncrossing step that, given an arbitrary set system with odd intersection, finds an odd set with value not more than the maximum of the initial system.