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. We note that graph and hypergraph cut functions are symmetric submodular and moreover, both monotone submodular functions and symmetric submodular functions are posimodular submodular. We analyze the approximation factor of Narayanan, Roy, and Patkar's algorithm to 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). 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：在子模$k$划分问题中，输入由一个定义在有限基集$V$上的非负子模函数$f$（通过评估预言机给出）以及一个正整数$k$组成，目标是找到基集$V$的一个划分为$k$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子模$k$划分的算法，并证明了对于图割函数的特例，其算法的近似因子为$2$（随后被Ravi和Sinha（Journal of Operational Research，2008）重新发现）。本文研究了该算法对三类子模函数族——单调子模、对称子模和正模子模——的近似因子。我们注意到，图割函数和超图割函数是对称子模的，并且单调子模函数和对称子模函数均属于正模子模函数。通过分析Narayanan、Roy和Patkar算法的近似因子，我们得到以下结果：1. 该算法对单调子模$k$划分的近似因子为$4/3$。该结果改进了其他算法可实现的$2$因子。此外，我们的上界$4/3$与最近在多项式次函数评估查询下限（Santiago，IWOCA 2021）中证明的下界相匹配。2. 该算法对对称子模$k$划分的近似因子为$2$。该结果将近似因子分析推广至图割函数之外。3. 该算法对正模子模$k$划分的近似因子为$2$。我们还构造了一个例子，表明该算法对任意子模函数的近似因子为$\Omega(n/k)$。