---
OA_place: publisher
OA_type: hybrid
_id: '21766'
abstract:
- lang: eng
  text: We provide a new characterisation of the decades old open problem of extending
    bilipschitz mappings given on a Euclidean separated net. In particular, this allows
    for the complete positive solution of the open problem in dimension two. Along
    the way, we develop a set of tools for bilipschitz extensions of mappings between
    subsets of Euclidean spaces.
acknowledgement: "The present work developed from a research visit of M.D. to V.K.
  at IST Austria, funded by\r\na London Mathematical Society Research in Pairs grant.
  This work was done while V.K. was fully funded by the Austria Science Fund (FWF)
  [M 3100-N]."
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Michael
  full_name: Dymond, Michael
  last_name: Dymond
- first_name: Vojtech
  full_name: Kaluza, Vojtech
  id: 21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E
  last_name: Kaluza
  orcid: 0000-0002-2512-8698
citation:
  ama: Dymond M, Kaluza V. Extending bilipschitz mappings between separated nets.
    <i>Annales Fennici Mathematici</i>. 2026;51(1):237-260. doi:<a href="https://doi.org/10.54330/afm.181562">10.54330/afm.181562</a>
  apa: Dymond, M., &#38; Kaluza, V. (2026). Extending bilipschitz mappings between
    separated nets. <i>Annales Fennici Mathematici</i>. Finnish Mathematical Society.
    <a href="https://doi.org/10.54330/afm.181562">https://doi.org/10.54330/afm.181562</a>
  chicago: Dymond, Michael, and Vojtech Kaluza. “Extending Bilipschitz Mappings between
    Separated Nets.” <i>Annales Fennici Mathematici</i>. Finnish Mathematical Society,
    2026. <a href="https://doi.org/10.54330/afm.181562">https://doi.org/10.54330/afm.181562</a>.
  ieee: M. Dymond and V. Kaluza, “Extending bilipschitz mappings between separated
    nets,” <i>Annales Fennici Mathematici</i>, vol. 51, no. 1. Finnish Mathematical
    Society, pp. 237–260, 2026.
  ista: Dymond M, Kaluza V. 2026. Extending bilipschitz mappings between separated
    nets. Annales Fennici Mathematici. 51(1), 237–260.
  mla: Dymond, Michael, and Vojtech Kaluza. “Extending Bilipschitz Mappings between
    Separated Nets.” <i>Annales Fennici Mathematici</i>, vol. 51, no. 1, Finnish Mathematical
    Society, 2026, pp. 237–60, doi:<a href="https://doi.org/10.54330/afm.181562">10.54330/afm.181562</a>.
  short: M. Dymond, V. Kaluza, Annales Fennici Mathematici 51 (2026) 237–260.
corr_author: '1'
date_created: 2026-04-26T22:01:47Z
date_published: 2026-04-17T00:00:00Z
date_updated: 2026-04-28T12:06:00Z
day: '17'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.54330/afm.181562
external_id:
  arxiv:
  - '2507.22007'
file:
- access_level: open_access
  checksum: 442023926a3803d5d6ca8db8dbc4af1c
  content_type: application/pdf
  creator: dernst
  date_created: 2026-04-28T12:03:13Z
  date_updated: 2026-04-28T12:03:13Z
  file_id: '21772'
  file_name: 2026_AnnalesFenniciMath_Dymond.pdf
  file_size: 342082
  relation: main_file
  success: 1
file_date_updated: 2026-04-28T12:03:13Z
has_accepted_license: '1'
intvolume: '        51'
issue: '1'
keyword:
- Lipschitz
- bilipschitz
- extension
- separated net.
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc/4.0/
month: '04'
oa: 1
oa_version: Published Version
page: 237-260
project:
- _id: fc35eaa2-9c52-11eb-aca3-88501ab155e9
  grant_number: M03100
  name: Spectra and topology of graphs and of simplicial complexes
publication: Annales Fennici Mathematici
publication_identifier:
  eissn:
  - 2737-114X
  issn:
  - 2737-0690
publication_status: published
publisher: Finnish Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: Extending bilipschitz mappings between separated nets
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  short: CC BY-NC (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 51
year: '2026'
...
---
OA_place: publisher
OA_type: hybrid
_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"
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).
article_processing_charge: Yes (via OA deal)
article_type: original
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>Mathematical Programming</i>. 2025;209:279-322. doi:<a href="https://doi.org/10.1007/s10107-024-02064-5">10.1007/s10107-024-02064-5</a>
  apa: Dvorak, M., &#38; Kolmogorov, V. (2025). Generalized minimum 0-extension problem
    and discrete convexity. <i>Mathematical Programming</i>. Springer Nature. <a href="https://doi.org/10.1007/s10107-024-02064-5">https://doi.org/10.1007/s10107-024-02064-5</a>
  chicago: Dvorak, Martin, and Vladimir Kolmogorov. “Generalized Minimum 0-Extension
    Problem and Discrete Convexity.” <i>Mathematical Programming</i>. Springer Nature,
    2025. <a href="https://doi.org/10.1007/s10107-024-02064-5">https://doi.org/10.1007/s10107-024-02064-5</a>.
  ieee: M. Dvorak and V. Kolmogorov, “Generalized minimum 0-extension problem and
    discrete convexity,” <i>Mathematical Programming</i>, vol. 209. Springer Nature,
    pp. 279–322, 2025.
  ista: Dvorak M, Kolmogorov V. 2025. Generalized minimum 0-extension problem and
    discrete convexity. Mathematical Programming. 209, 279–322.
  mla: Dvorak, Martin, and Vladimir Kolmogorov. “Generalized Minimum 0-Extension Problem
    and Discrete Convexity.” <i>Mathematical Programming</i>, vol. 209, Springer Nature,
    2025, pp. 279–322, doi:<a href="https://doi.org/10.1007/s10107-024-02064-5">10.1007/s10107-024-02064-5</a>.
  short: M. Dvorak, V. Kolmogorov, Mathematical Programming 209 (2025) 279–322.
corr_author: '1'
date_created: 2021-09-27T10:48:23Z
date_published: 2025-01-01T00:00:00Z
date_updated: 2025-05-19T13:52:10Z
day: '01'
ddc:
- '004'
department:
- _id: GradSch
- _id: VlKo
doi: 10.1007/s10107-024-02064-5
external_id:
  arxiv:
  - '2109.10203'
  isi:
  - '001176563300001'
file:
- access_level: open_access
  checksum: 25d9bd490719b45eca84f4d93a06c69f
  content_type: application/pdf
  creator: dernst
  date_created: 2025-04-16T09:36:08Z
  date_updated: 2025-04-16T09:36:08Z
  file_id: '19578'
  file_name: 2025_MathProgramming_Dvorak.pdf
  file_size: 839510
  relation: main_file
  success: 1
file_date_updated: 2025-04-16T09:36:08Z
has_accepted_license: '1'
intvolume: '       209'
isi: 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
license: https://creativecommons.org/licenses/by/4.0/
month: '01'
oa: 1
oa_version: Published Version
page: 279-322
publication: Mathematical Programming
publication_identifier:
  eissn:
  - 1436-4646
  issn:
  - 0025-5610
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Generalized minimum 0-extension problem and discrete convexity
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 209
year: '2025'
...