---
_id: '10045'
abstract:
- lang: eng
  text: "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.\r\n"
article_number: '2109.10203'
article_processing_charge: No
arxiv: 1
author:
- first_name: Martin
  full_name: Dvorak, Martin
  id: 40ED02A8-C8B4-11E9-A9C0-453BE6697425
  last_name: Dvorak
  orcid: 0000-0001-5293-214X
- first_name: Vladimir
  full_name: Kolmogorov, Vladimir
  id: 3D50B0BA-F248-11E8-B48F-1D18A9856A87
  last_name: Kolmogorov
citation:
  ama: Dvorak M, Kolmogorov V. Generalized minimum 0-extension problem and discrete
    convexity. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2109.10203">10.48550/arXiv.2109.10203</a>
  apa: Dvorak, M., &#38; Kolmogorov, V. (n.d.). Generalized minimum 0-extension problem
    and discrete convexity. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2109.10203">https://doi.org/10.48550/arXiv.2109.10203</a>
  chicago: Dvorak, Martin, and Vladimir Kolmogorov. “Generalized Minimum 0-Extension
    Problem and Discrete Convexity.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2109.10203">https://doi.org/10.48550/arXiv.2109.10203</a>.
  ieee: M. Dvorak and V. Kolmogorov, “Generalized minimum 0-extension problem and
    discrete convexity,” <i>arXiv</i>. .
  ista: Dvorak M, Kolmogorov V. Generalized minimum 0-extension problem and discrete
    convexity. arXiv, 2109.10203.
  mla: Dvorak, Martin, and Vladimir Kolmogorov. “Generalized Minimum 0-Extension Problem
    and Discrete Convexity.” <i>ArXiv</i>, 2109.10203, doi:<a href="https://doi.org/10.48550/arXiv.2109.10203">10.48550/arXiv.2109.10203</a>.
  short: M. Dvorak, V. Kolmogorov, ArXiv (n.d.).
date_created: 2021-09-27T10:48:23Z
date_published: 2021-09-21T00:00:00Z
date_updated: 2023-05-03T10:40:16Z
day: '21'
ddc:
- '004'
department:
- _id: GradSch
- _id: VlKo
doi: 10.48550/arXiv.2109.10203
external_id:
  arxiv:
  - '2109.10203'
file:
- access_level: open_access
  checksum: e7e83065f7bc18b9c188bf93b5ca5db6
  content_type: application/pdf
  creator: mdvorak
  date_created: 2021-09-27T10:54:51Z
  date_updated: 2021-09-27T10:54:51Z
  file_id: '10046'
  file_name: Generalized-0-Ext.pdf
  file_size: 603672
  relation: main_file
  success: 1
file_date_updated: 2021-09-27T10:54:51Z
has_accepted_license: '1'
keyword:
- minimum 0-extension problem
- metric labeling problem
- discrete metric spaces
- metric extensions
- computational complexity
- valued constraint satisfaction problems
- discrete convex analysis
- L-convex functions
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.2109.10203'
month: '09'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: submitted
status: public
title: Generalized minimum 0-extension problem and discrete convexity
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...