This paper studies the computational complexity of a robust variant of a two-stage submodular minimization problem that we call Robust Submodular Minimizer. In this problem, we are given $k$ submodular functions $f_1,\dots,f_k$ over a set family $2^V$, which represent $k$ possible scenarios in the future when we will need to find an optimal solution for one of these scenarios, i.e., a minimizer for one of the functions. The present task is to find a set $X \subseteq V$ that is close to some optimal solution for each $f_i$ in the sense that some minimizer of $f_i$ can be obtained from $X$ by adding/removing at most $d$ elements for a given integer $d$. The main contribution of this paper is to provide a complete computational map of this problem with respect to parameters $k$ and $d$, which reveals a tight complexity threshold for both parameters: (1) Robust Submodular Minimizer can be solved in polynomial time when $k \leq 2$, but is NP-hard if $k$ is a constant with $k \geq 3$. (2) Robust Submodular Minimizer can be solved in polynomial time when $d=0$, but is NP-hard if $d$ is a constant with $d \geq 1$. (3) Robust Submodular Minimizer is fixed-parameter tractable when parameterized by $(k,d)$. We also show that if some submodular function $f_i$ has a polynomial number of minimizers, then the problem becomes fixed-parameter tractable when parameterized by $d$. We remark that all our hardness results hold even if each submodular function is given by a cut function of a directed graph.

翻译：本文研究一类鲁棒性两阶段子模最小化问题的计算复杂度,我们称之为鲁棒子模最小化器。在该问题中,给定定义在集合族$2^V$上的$k$个子模函数$f_1,\dots,f_k$,这些函数代表未来可能出现的$k$种场景,每种场景下需要寻找对应函数的最优解(即最小化器)。当前任务是找到一个子集$X \subseteq V$,使其与每个$f_i$的某个最优解保持"接近",具体而言:存在某个$f_i$的最小化器,可通过从$X$中最多添加或删除$d$个元素得到(其中$d$为给定整数)。本文的主要贡献在于揭示了该问题关于参数$k$和$d$的完整计算复杂度图谱,并给出了两个参数的严格阈值：(1)当$k \leq 2$时,鲁棒子模最小化器可在多项式时间内求解;但当$k \geq 3$为常数时,问题为NP难。(2)当$d=0$时,鲁棒子模最小化器可在多项式时间内求解;但当$d \geq 1$为常数时,问题为NP难。(3)当以$(k,d)$为参数时,鲁棒子模最小化器是固定参数可解的。进一步地,我们证明若某子模函数$f_i$具有多项式数量的最小化器,则该问题关于参数$d$是固定参数可解的。需要指出的是,本文所有难度结果即使在每个子模函数均由有向图的割函数表示时依然成立。