In submodular multiway partition (SUB-MP), the input is a non-negative submodular function $f:2^V \rightarrow \mathbb{R}_{\ge 0}$ given by an evaluation oracle along with $k$ terminals $t_1, t_2, \ldots, t_k\in V$. The goal is to find a partition $V_1, V_2, \ldots, V_k$ of $V$ with $t_i\in V_i$ for every $i\in [k]$ in order to minimize $\sum_{i=1}^k f(V_i)$. In this work, we focus on SUB-MP when the input function is monotone (termed MONO-SUB-MP). MONO-SUB-MP formulates partitioning problems over several interesting structures -- e.g., matrices, matroids, graphs, and hypergraphs. MONO-SUB-MP is NP-hard since the graph multiway cut problem can be cast as a special case. We investigate the approximability of MONO-SUB-MP: we show that it admits a $4/3$-approximation and does not admit a $(10/9-\epsilon)$-approximation for every constant $\epsilon>0$. Next, we study a special case of MONO-SUB-MP where the monotone submodular function of interest is the coverage function of an input graph, termed GRAPH-COVERAGE-MP. GRAPH-COVERAGE-MP is equivalent to the classic multiway cut problem for the purposes of exact optimization. We show that GRAPH-COVERAGE-MP admits a $1.125$-approximation and does not admit a $(1.00074-\epsilon)$-approximation for every constant $\epsilon>0$ assuming the Unique Games Conjecture. These results separate GRAPH-COVERAGE-MP from graph multiway cut in terms of approximability.

翻译：在次模多路划分（SUB-MP）问题中，输入为一个由评估预言机给出的非负次模函数 $f:2^V \rightarrow \mathbb{R}_{\ge 0}$ 以及 $k$ 个终端节点 $t_1, t_2, \ldots, t_k\in V$。目标是找到 $V$ 的一个划分 $V_1, V_2, \ldots, V_k$，使得对于每个 $i\in [k]$ 有 $t_i\in V_i$，并最小化 $\sum_{i=1}^k f(V_i)$。本文聚焦于输入函数为单调次模函数时的 SUB-MP 问题（称为 MONO-SUB-MP）。MONO-SUB-MP 可建模多种重要结构上的划分问题——例如矩阵、拟阵、图与超图。由于图多路割问题可视为其特例，MONO-SUB-MP 是 NP 难问题。我们研究了 MONO-SUB-MP 的近似性：证明其存在 $4/3$ 近似算法，且对于任意常数 $\epsilon>0$ 不存在 $(10/9-\epsilon)$ 近似算法。随后，我们研究了 MONO-SUB-MP 的一个特例，其中单调次模函数为输入图的覆盖函数，称为 GRAPH-COVERAGE-MP。在精确优化意义上，GRAPH-COVERAGE-MP 等价于经典的多路割问题。我们证明：在唯一游戏猜想成立的前提下，GRAPH-COVERAGE-MP 存在 $1.125$ 近似算法，且对于任意常数 $\epsilon>0$ 不存在 $(1.00074-\epsilon)$ 近似算法。这些结果在近似性层面将 GRAPH-COVERAGE-MP 与图多路割问题区分开来。