TY - GEN

T1 - Cospanning Characterizations of Violator and Co-violator Spaces

AU - Kempner, Yulia

AU - Levit, Vadim E.

N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

PY - 2024

Y1 - 2024

N2 - Given a finite set E and an operator σ:2E⟶2E, 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: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.

AB - Given a finite set E and an operator σ:2E⟶2E, 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: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.

KW - Co-violator space

KW - Cospanning relation

KW - Uniquely generated violator space

UR - http://www.scopus.com/inward/record.url?scp=85197879534&partnerID=8YFLogxK

U2 - 10.1007/978-3-031-52969-6_11

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

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AN - SCOPUS:85197879534

SN - 9783031529689

T3 - Springer Proceedings in Mathematics and Statistics

SP - 109

EP - 117

BT - Combinatorics, Graph Theory and Computing - SEICCGTC 2021

A2 - Hoffman, Frederick

A2 - Holliday, Sarah

A2 - Rosen, Zvi

A2 - Shahrokhi, Farhad

A2 - Wierman, John

T2 - 52nd Southeastern International Conference on Combinatorics, Graph Theory and Computing, SEICCGTC 2021

Y2 - 8 March 2021 through 12 March 2021

ER -