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# Parameterized Complexity of Submodular Minimization Under Uncertainty

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LIPIcs.SWAT.2024.30.pdf
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## Acknowledgements

The authors are grateful to the organizers of the 14th Emléktábla Workshop which took place in Vác, Hungary in 2023, and provided a great opportunity for our initial discussions.

## Cite As

Naonori Kakimura and Ildikó Schlotter. Parameterized Complexity of Submodular Minimization Under Uncertainty. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.30

## 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,… ,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 ∈ 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: - 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). 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Submodular optimization and polymatroids
• Theory of computation → Fixed parameter tractability
• Mathematics of computing → Combinatorial optimization
##### Keywords
• Submodular minimization
• optimization under uncertainty
• parameterized complexity
• cut function

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