Cospanning Characterizations of Violator and Co-violator Spaces

Yulia Kempner; Vadim E. Levit

doi:10.1007/978-3-031-52969-6_11

Cospanning Characterizations of Violator and Co-violator Spaces

Yulia Kempner, Vadim E. Levit

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Given a finite set E and an operator σ:2^E⟶2^E, two subsets X,Y⊆E are cospanning if σ(X)=σ(Y) (Korte, Lovász, Schrader; 1991). We investigate cospanning relations on violator spaces. A notion of a violator space was introduced in (Gärtner, Matoušek, Rüst, Škovroňby; 2008) as a combinatorial framework that encompasses linear programming and other geometric optimization problems. Violator spaces are defined by violator operators. We introduce co-violator spaces based on contracting operators known also as choice functions. Let α,β:2^E⟶2^E be a violator operator and a co-violator operator, respectively. Cospanning characterizations of violator spaces allow us to obtain some new properties of violator operators and co-violator operators, emphasizing their interconnections. In particular, we show that uniquely generated violator spaces satisfy so-called Krein-Milman properties, i.e., α(βX)=α(X) and βαX=βX for every X⊆E.

Original language	English
Title of host publication	Combinatorics, Graph Theory and Computing - SEICCGTC 2021
Editors	Frederick Hoffman, Sarah Holliday, Zvi Rosen, Farhad Shahrokhi, John Wierman
Publisher	Springer
Pages	109-117
Number of pages	9
ISBN (Print)	9783031529689
DOIs	https://doi.org/10.1007/978-3-031-52969-6_11
State	Published - 2024
Externally published	Yes
Event	52nd Southeastern International Conference on Combinatorics, Graph Theory and Computing, SEICCGTC 2021 - Boca Raton, United States Duration: 8 Mar 2021 → 12 Mar 2021

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	448
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	52nd Southeastern International Conference on Combinatorics, Graph Theory and Computing, SEICCGTC 2021
Country/Territory	United States
City	Boca Raton
Period	8/03/21 → 12/03/21

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

Keywords

Co-violator space
Cospanning relation
Uniquely generated violator space

Access to Document

10.1007/978-3-031-52969-6_11

Cite this

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title = "Cospanning Characterizations of Violator and Co-violator Spaces",

abstract = "Given a finite set E and an operator σ:2E⟶2E, two subsets X,Y⊆E are cospanning if σ(X)=σ(Y) (Korte, Lov{\'a}sz, Schrader; 1991). We investigate cospanning relations on violator spaces. A notion of a violator space was introduced in (G{\"a}rtner, Matou{\v s}ek, R{\"u}st, {\v S}kovro{\v n}by; 2008) as a combinatorial framework that encompasses linear programming and other geometric optimization problems. Violator spaces are defined by violator operators. We introduce co-violator spaces based on contracting operators known also as choice functions. Let α,β:2E⟶2E be a violator operator and a co-violator operator, respectively. Cospanning characterizations of violator spaces allow us to obtain some new properties of violator operators and co-violator operators, emphasizing their interconnections. In particular, we show that uniquely generated violator spaces satisfy so-called Krein-Milman properties, i.e., α(βX)=α(X) and βαX=βX for every X⊆E.",

keywords = "Co-violator space, Cospanning relation, Uniquely generated violator space",

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Kempner, Y & Levit, VE 2024, Cospanning Characterizations of Violator and Co-violator Spaces. in F Hoffman, S Holliday, Z Rosen, F Shahrokhi & J Wierman (eds), Combinatorics, Graph Theory and Computing - SEICCGTC 2021. Springer Proceedings in Mathematics and Statistics, vol. 448, Springer, pp. 109-117, 52nd Southeastern International Conference on Combinatorics, Graph Theory and Computing, SEICCGTC 2021, Boca Raton, United States, 8/03/21. https://doi.org/10.1007/978-3-031-52969-6_11

Cospanning Characterizations of Violator and Co-violator Spaces. / Kempner, Yulia; Levit, Vadim E.
Combinatorics, Graph Theory and Computing - SEICCGTC 2021. ed. / Frederick Hoffman; Sarah Holliday; Zvi Rosen; Farhad Shahrokhi; John Wierman. Springer, 2024. p. 109-117 (Springer Proceedings in Mathematics and Statistics; Vol. 448).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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