New Approximations and Hardness Results for Submodular Partitioning Problems

We consider the following class of submodular k-multiway partitioning problems: (Sub-$k$-MP) $\min \sum_{i=1}^k f(S_i): S_1 \uplus S_2 \uplus \cdots \uplus S_k = V \mbox{ and } S_i \neq \emptyset \mbox{ for all }i\in [k]$. Here $f$ is a non-negative submodular function, and $\uplus$ denotes the union of disjoint sets. Hence the goal is to partition $V$ into $k$ non-empty sets $S_1,S_2,\ldots,S_k$ such that $\sum_{i=1}^k f(S_i)$ is minimized. These problems were introduced by Zhao et al. partly motivated by applications to network reliability analysis, VLSI design, hypergraph cut, and other partitioning problems. In this work we revisit this class of problems and shed some light onto their hardness of approximation in the value oracle model. We provide new unconditional hardness results for Sub-$k$-MP in the special settings where the function $f$ is either monotone or symmetric. For symmetric functions we show that given any $\epsilon > 0$, any algorithm achieving a $(2 - \epsilon)$-approximation requires exponentially many queries in the value oracle model. For monotone objectives we show that given any $\epsilon > 0$, any algorithm achieving a $(4/3 - \epsilon)$-approximation requires exponentially many queries in the value oracle model. We then extend Sub-$k$-MP to a larger class of partitioning problems, where the functions $f_i(S_i)$ can now be all different, and there is a more general partitioning constraint $ S_1 \uplus S_2 \uplus \cdots \uplus S_k \in \mathcal{F}$ for some family $\mathcal{F} \subseteq 2^V$ of feasible sets. We provide a black box reduction that allows us to leverage several existing results from the literature; leading to new approximations for this class of problems.