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 $k$-分割问题中，输入定义在有限基集 $V$ 上的非负次模函数 $f$（由评估预言机给出）和正整数 $k$，目标是找到基集 $V$ 的一个划分，将其分为 $k$ 个非空子集 $V_1, V_2, ..., V_k$，以最小化 $\sum_{i=1}^k f(V_i)$。Narayanan、Roy 和 Patkar（Journal of Algorithms, 1996）基于主划分序列设计了次模 $k$-分割算法，并证明在图割函数这一特例下（后被 Ravi 和 Sinha（Journal of Operational Research, 2008）重新发现），其算法的近似因子为 $2$。本文研究该算法对三类次模函数子族（单调次模、对称次模和正模函数）的近似因子，并得到以下结果：1. 对于单调次模 $k$-分割，算法的近似因子为 $4/3$。该结果改进了其他算法可实现的 $2$-近似因子。此外，我们的 $4/3$ 上界与最近在多项式次函数评估查询次数下证明的下界（Santiago, IWOCA 2021）相匹配。该 $4/3$ 上界也是针对某个特殊单调次模 $k$-分割的图划分问题首个优于 $2$ 的改进。2. 对于对称次模 $k$-分割，算法的近似因子为 $2$。该结果将近似因子分析推广至图割函数之外。3. 对于正模次模 $k$-分割，算法的近似因子为 $2$。我们进一步构造反例，表明该算法对任意次模函数的近似因子为 $\Omega(n/k)$。