Robust Optimization for Submodular Minimization under Uncertainty

Temel Kavramlar

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.

Özet

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:

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).
4. 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.

İstatistikler

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Alıntılar

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Önemli Bilgiler Şuradan Elde Edildi

Parameterized Complexity of Submodular Minimization under Uncertainty

by Naon... : arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07516.pdf

Daha Derin Sorular

How can the techniques developed in this paper be applied to other robust optimization problems beyond submodular minimization

The techniques developed in this paper for the Robust Submodular Minimizer problem can be applied to other robust optimization problems by adapting the framework to different objective functions and constraints. The key idea is to model uncertainty in the optimization process by considering multiple scenarios and finding solutions that are robust across these scenarios. This approach can be extended to various optimization problems in different domains such as logistics, finance, and machine learning. By formulating the problem as a two-stage optimization with uncertainty, similar to the Robust Submodular Minimizer, one can analyze the computational complexity and design efficient algorithms for finding robust solutions.

What are some real-world applications of the Robust Submodular Minimizer problem and how can the insights from this paper inform the design of practical algorithms for those applications

Real-world applications of the Robust Submodular Minimizer problem can be found in various fields such as supply chain management, network design, and resource allocation. For example, in supply chain management, the problem can be used to optimize inventory levels under uncertain demand scenarios. Insights from this paper, such as the fixed-parameter tractability results and the use of compact representations for submodular functions, can inform the design of practical algorithms for these applications. By leveraging the parameterized complexity analysis and the bounded search-tree technique, efficient algorithms can be developed to handle uncertainty and variability in real-world optimization problems.

Is there a deeper connection between the complexity of the Robust Submodular Minimizer problem and the structure of the underlying submodular functions beyond the number of minimizers

The complexity of the Robust Submodular Minimizer problem is closely related to the structure of the underlying submodular functions, particularly the number of minimizers and the interactions between them. The presence of a polynomial number of minimizers for a submodular function can lead to fixed-parameter tractability results, as shown in the paper. Understanding the properties of submodular functions, such as the distributive lattice formed by the minimizers, can provide insights into the computational complexity of the problem. Deeper connections between the complexity of the problem and the structural properties of submodular functions can lead to further advancements in algorithm design and optimization theory.