Polynomial algorithm for $k$-partition minimization of monotone submodular function

For a fixed $k$, this study considers $k$-partition minimization of submodular system $(V,f)$ with a finite set $V$ and symmetric submodular function $f:2^V\mapsto \mathbb{R}$. Our algorithm uses the Queyranne’s (1998) algorithm for 2-partition minimization which arises at each step of the recursive decomposition of subsets of the original $k$-partition minimization. We show that the computational complexity of this minimizer is $O(n^{3(k-1)})$.