A polynomial algorithm for the k-cut problem for fixed k
Mathematics of Operations Research
Fast approximation algorithms for fractional packing and covering problems
Mathematics of Operations Research
A new approach to the minimum cut problem
Journal of the ACM (JACM)
Randomized rounding without solving the linear program
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Minimum cuts in near-linear time
Journal of the ACM (JACM)
Tree packing and approximating k-cuts
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Combinatorica
A Deterministic Algorithm for Finding All Minimum $k$-Way Cuts
SIAM Journal on Computing
Divide-and-Conquer Algorithms for Partitioning Hypergraphs and Submodular Systems
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Finding minimum 3-way cuts in hypergraphs
Information Processing Letters
Computing minimum multiway cuts in hypergraphs from hypertree packings
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Clique cover and graph separation: new incompressibility results
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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We present a simple and fast deterministic algorithm for the minimum k-way cut problem in a capacitated graph, that is, finding a set of edges with minimum total capacity whose removal splits the graph into at least k components. The algorithm packs O(mk3 log n) trees. Each new tree is a minimal spanning tree with respect to the edge utilizations, and the utilization of an edge is the number of times it has been used in previous spanning trees divided by its capacity. We prove that each minimum k-way cut is crossed at most 2k-2 times by one of the trees. We can enumerate all such cuts in ~O(n2k) time, which is hence the running time of our algorithm producing all minimum k-way cuts. The previous fastest deterministic algorithm of Kamidoi et al. [SICOMP'06] took O(n(4+o(1))k) time, so this is a near-quadratic improvement. Moreover, we essentially match the O(n(2-o(1))k) running time of the Monto Carlo (no correctness guarantee) randomized algorithm of Karger and Stein [JACM'96].