Keskeiset käsitteet
The core message of this paper is to provide a complete computational complexity map of the Robust Submodular Minimizer problem, which aims to find a set that is close to some optimal solution for each of the k given submodular functions, under a given recovery bound d.
Tiivistelmä
The paper studies the computational complexity of the Robust Submodular Minimizer problem, which is a two-stage robust optimization problem under uncertainty. The problem is defined as follows:
- There are k submodular functions f1, ..., fk over a set family 2^V, representing k possible scenarios in the future.
- The task is to find a set X ⊆ V that is close to some optimal solution for each fi, in the sense that some minimizer of fi can be obtained from X by adding/removing at most d elements.
The main contributions of the paper are:
- Robust Submodular Minimizer can be solved in polynomial time when k ≤ 2, but is NP-hard if k is a constant with k ≥ 3.
- Robust Submodular Minimizer can be solved in polynomial time when d = 0, but is NP-hard if d is a constant with d ≥ 1.
- Robust Submodular Minimizer is fixed-parameter tractable when parameterized by (k, d).
- If some submodular function fi has a polynomial number of minimizers, then the problem becomes fixed-parameter tractable when parameterized by d.
The authors use Birkhoff's representation theorem on distributive lattices to maintain the family of minimizers of a submodular function in a compact way, which allows them to solve the problem efficiently in certain cases. They also provide hardness results by reductions from an intermediate problem.