Parameterized Complexity of Submodular Minimization under Uncertainty
arxiv(2024)
摘要
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,…,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 ⊆ 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 ≤ 2, but is NP-hard if k is a
constant with k ≥ 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 ≥
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.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要