In submodular $k$-partition, the input is a non-negative submodular function $f$ defined over a finite ground set $V$ (given by an evaluation oracle) along with a positive integer $k$ and the goal is to find a partition of the ground set $V$ into $k$ non-empty parts $V_1, V_2, ..., V_k$ in order to minimize $\sum_{i=1}^k f(V_i)$. Narayanan, Roy, and Patkar (Journal of Algorithms, 1996) designed an algorithm for submodular $k$-partition based on the principal partition sequence and showed that the approximation factor of their algorithm is $2$ for the special case of graph cut functions (subsequently rediscovered by Ravi and Sinha (Journal of Operational Research, 2008)). In this work, we study the approximation factor of their algorithm for three subfamilies of submodular functions -- monotone, symmetric, and posimodular, and show the following results: 1. The approximation factor of their algorithm for monotone submodular $k$-partition is $4/3$. This result improves on the $2$-factor achievable via other algorithms. Moreover, our upper bound of $4/3$ matches the recently shown lower bound under polynomial number of function evaluation queries (Santiago, IWOCA 2021). Our upper bound of $4/3$ is also the first improvement beyond $2$ for a certain graph partitioning problem that is a special case of monotone submodular $k$-partition. 2. The approximation factor of their algorithm for symmetric submodular $k$-partition is $2$. This result generalizes their approximation factor analysis beyond graph cut functions. 3. The approximation factor of their algorithm for posimodular submodular $k$-partition is $2$. We also construct an example to show that the approximation factor of their algorithm for arbitrary submodular functions is $\Omega(n/k)$.

翻译：在次模 $k$-划分问题中，输入是一个定义在有限基集 $V$ 上的非负次模函数 $f$（通过评估预言机给出）以及一个正整数 $k$，目标是找到基集 $V$ 的一个划分为 $k$ 个非空子集 $V_1, V_2, \ldots, V_k$，使得 $\sum_{i=1}^k f(V_i)$ 最小化。Narayanan、Roy 和 Patkar（Journal of Algorithms, 1996）基于主划分序列设计了一种次模 $k$-划分算法，并证明该算法对于图割函数这一特例的近似比为 $2$（后被 Ravi 和 Sinha（Journal of Operational Research, 2008）重新发现）。本文研究了该算法对三类次模函数子族——单调次模、对称次模和正模次模——的近似比，并得到以下结果：1. 该算法对单调次模 $k$-划分的近似比为 $4/3$。这一结果优于其他算法可达到的 $2$ 倍近似比。此外，我们的上界 $4/3$ 与最近在多项式次数的函数评估查询下证明的下界（Santiago, IWOCA 2021）相匹配。$4/3$ 的上界也是首次将某个图划分问题（作为单调次模 $k$-划分的特例）的近似比改进到 $2$ 以下。2. 该算法对对称次模 $k$-划分的近似比为 $2$。这一结果将其近似比分析推广至图割函数之外。3. 该算法对正模次模 $k$-划分的近似比为 $2$。我们还构造了一个例子，证明该算法对任意次模函数的近似比为 $\Omega(n/k)$。