Parameterized Complexity of Submodular Minimization under Uncertainty

TL;DR

The paper addresses Robust Submodular Minimizer, a two-stage robust optimization problem over submodular minimizers: given $k$ submodular functions $f_1,\dots,f_k$ on $2^V$, find $X\subseteq V$ such that for each $i$ there exists $Y_i\in\arg\min f_i$ with $|X\triangle Y_i|\le d$. It develops a complete parameterized complexity map with respect to $(k,d)$ by leveraging Birkhoff's representation to compactly encode minimizers as a DAG and reducing the problem to a Multi-Budgeted Directed Cut framework, yielding a polynomial-time solution for $k=2$ and an FPT algorithm for general $(k,d)$; for $k=2$ the construction reduces to a minimum $(s,t)$-cut on a small graph. The results include NP-hardness for $k\ge 3$ or $d\ge 1$, W[1]-hardness in $k$ even when each $\arg\min f_i$ is polynomial-sized, and fixed-parameter tractability in $d$ when some $|\arg\min f_i|$ is polynomially bounded, offering a nuanced landscape of tractability. When some submodular function has a polynomial number of minimizers, the problem becomes FPT in $d$, connecting robust submodular minimization to directed-cut techniques and providing a comprehensive complexity map for this robustness setting.

Abstract

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 \emph{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 $d \in \mathbb{N}$. 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 $d$. On the other hand, the problem remains $\mathsf{W}[1]$-hard parameterized by $k$ even if each function $f_i$ has at most~$|V|$ minimizers. We remark that all our hardness results hold even if each submodular function is given by a cut function of a directed graph.

Theorems & Definitions (1)

• Theorem 1: Birkhoff's representation theorem Birkhoff37