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.

中文翻译：

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子模划分问题的新近似值和硬度结果

我们考虑以下类别的子模 k-multiway 分区问题： (Sub-$k$-MP) $\min \sum_{i=1}^kf(S_i): S_1 \uplus S_2 \uplus \cdots \uplus S_k = V \mbox{ 和 } S_i \neq \emptyset \mbox{ 对于所有的 }i\in [k]$。这里 $f$ 是一个非负子模函数，$\uplus$ 表示不相交集的并集。因此，目标是将 $V$ 划分为 $k$ 非空集 $S_1,S_2,\ldots,S_k$，使得 $\sum_{i=1}^kf(S_i)$ 最小化。这些问题是由 Zhao 等人提出的。部分原因是网络可靠性分析、VLSI 设计、超图切割和其他分区问题的应用。在这项工作中，我们重新审视了这类问题，并阐明了它们在价值预言模型中的逼近难度。我们在函数 $f$ 是单调或对称的特殊设置中为 Sub-$k$-MP 提供新的无条件硬度结果。对于对称函数，我们表明，给定任何 $\epsilon > 0$，任何实现 $(2 - \epsilon)$ 近似的算法都需要在价值预言模型中以指数方式进行多次查询。对于单调目标，我们表明，给定任何 $\epsilon > 0$，任何实现 $(4/3 - \epsilon)$ 近似的算法都需要在价值预言模型中以指数方式进行多次查询。然后我们将 Sub-$k$-MP 扩展到更大的分区问题，其中函数 $f_i(S_i)$ 现在可以完全不同，并且有一个更一般的分区约束 $ S_1 \uplus S_2 \uplus \cdots \uplus S_k \in \mathcal{F}$ 对于某些家庭 $\mathcal{F} \subseteq 2^V$ 的可行集。我们提供了一个黑盒减少，使我们能够利用文献中的几个现有结果；导致此类问题的新近似值。