A Faster Algorithm for Computing Minimum 5-Way and 6-Way Cuts in Graphs

Abstract

For an edge-weighted graph G with n vertices and m edges, the minimum k-way cut problem is to find a partition of the vertex set into k non-empty subsets so that the weight sum of edges between different subsets is minimized. For this problem with k = 5 and 6, we present a deterministic algorithm that runs in $O\left( {n^{k - 2} \left( {nF\left( {n,m} \right) + C_2 \left( {n,m} \right) + n^2 } \right)} \right) = O\left( {mn^k \log \left( {n^2 /m} \right)} \right)$ time, 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 bounds $\tilde O\left( {mn^5 } \right)$ for k = 5 and $\tilde O\left( {mn^6 } \right)$ for k = 6 improve the previous best randomized bounds $\tilde O\left( {n^8 } \right)$ and $\tilde O\left( {n^{10} } \right)$, respectively.

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.