# Generalized minimum 0-extension problem and discrete convexity

Dvorak M, Kolmogorov V. 2024. Generalized minimum 0-extension problem and discrete convexity. Mathematical Programming.

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https://doi.org/10.1007/s10107-024-02064-5
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Abstract

Given a fixed finite metric space (V,μ), the {\em minimum 0-extension problem}, denoted as 0-Ext[μ], is equivalent to the following optimization problem: minimize function of the form minx∈Vn∑ifi(xi)+∑ijcijμ(xi,xj) where cij,cvi are given nonnegative costs and fi:V→R are functions given by fi(xi)=∑v∈Vcviμ(xi,v). The computational complexity of 0-Ext[μ] has been recently established by Karzanov and by Hirai: if metric μ is {\em orientable modular} then 0-Ext[μ] can be solved in polynomial time, otherwise 0-Ext[μ] is NP-hard. To prove the tractability part, Hirai developed a theory of discrete convex functions on orientable modular graphs generalizing several known classes of functions in discrete convex analysis, such as L♮-convex functions. We consider a more general version of the problem in which unary functions fi(xi) can additionally have terms of the form cuv;iμ(xi,{u,v}) for {u,v}∈F, where set F⊆(V2) is fixed. We extend the complexity classification above by providing an explicit condition on (μ,F) for the problem to be tractable. In order to prove the tractability part, we generalize Hirai's theory and define a larger class of discrete convex functions. It covers, in particular, another well-known class of functions, namely submodular functions on an integer lattice. Finally, we improve the complexity of Hirai's algorithm for solving 0-Ext on orientable modular graphs.

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Date Published

2024-03-07

Journal Title

Mathematical Programming

Acknowledgement

We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions. Open access funding provided by Institute of Science and Technology (IST Austria).

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IST-REx-ID

### Cite this

Dvorak M, Kolmogorov V. Generalized minimum 0-extension problem and discrete convexity.

*Mathematical Programming*. 2024. doi:10.1007/s10107-024-02064-5Dvorak, M., & Kolmogorov, V. (2024). Generalized minimum 0-extension problem and discrete convexity.

*Mathematical Programming*. Springer Nature. https://doi.org/10.1007/s10107-024-02064-5Dvorak, Martin, and Vladimir Kolmogorov. “Generalized Minimum 0-Extension Problem and Discrete Convexity.”

*Mathematical Programming*. Springer Nature, 2024. https://doi.org/10.1007/s10107-024-02064-5.M. Dvorak and V. Kolmogorov, “Generalized minimum 0-extension problem and discrete convexity,”

*Mathematical Programming*. Springer Nature, 2024.Dvorak, Martin, and Vladimir Kolmogorov. “Generalized Minimum 0-Extension Problem and Discrete Convexity.”

*Mathematical Programming*, Springer Nature, 2024, doi:10.1007/s10107-024-02064-5.**All files available under the following license(s):**

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arXiv 2109.10203