Abstract
For an edge-weighted graph G with n vertices and m edges, we present a new deterministic algorithm for computing a minimum k-way cut for k = 3, 4. The algorithm runs in O(n k−2(nF(n,m) + C 2(n,m) + n 2)) = O(mn k log(n 2/m)) time for k = 3, 4, where F(n,m) and C 2(n,m) denote respectively the time bounds required to solve the maximum flow problem and the minimum 2-way cut problem in G. The bound for k = 3 matches the current best deterministic bound Õ (mn 3) for weighted graphs, but improves the bound Õ(mn 3) to O(n(nF(n,m) + C 2(n,m) + n 2)) = O(min{mn 8/3,m 3/2 n 2}) for unweighted graphs. The bound Õ(mn 4) for k = 4 improves the previous best randomized bound Õ(n 6) (for m = o(n 2)). The algorithm is then generalized to the problem of finding a minimum 3-way cut in a symmetric submodular system.
This research was partially supported by the Scientific Grant-in-Aid from Ministry of Education, Science, Sports and Culture of Japan, and the subsidy from the Inamori Foundation.
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Nagamochi, H., Ibaraki, T. (1999). A Fast Algorithm for Computing Minimum 3-Way and 4-Way Cuts. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1999. Lecture Notes in Computer Science, vol 1610. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48777-8_28
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