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Independence Structures and Submodular Functions

Independence Structures and Submodular Functions INDEPENDENCE STRUCTURES AND SUBMODULAR FUNCTIONS C. J. H. McDIARMID 1. Preliminaries We prove that a submodular function (not necessarily non-decreasing) induces in a natural way a pre-independence structure, thus extending known results [1, 3]. For any set A, the collection of subsets of A is denoted by 2 , and the cardinal of A is denoted by \A\. All functions, including |. |, are assumed to take values from {0, 1, ..., oo}. For sets A, B we write A c c B if A is a finite subse t of B. Given a set S, a non-empty set $ £ 2 is a pre-independence structure on S if it satisfies (1) ifXeS and 7 £ X, then 7e^ ; (2) if X, Y e<£ are finite and | Y\ = \X\ +1, then there exists y e Y\X such that X\j{y}<=£. We say that & is of finite character if it satisfies (3) if X c S and Y e<£ for every YccI , then Ze<?. An independence structure is a pre-independence structure of finite character. The rank function p of a pre-independence structure <f on S is defined by pX = sup{|Y|: YeS, YcczX} (XgS). Given a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Independence Structures and Submodular Functions

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/5.1.18
Publisher site
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Abstract

INDEPENDENCE STRUCTURES AND SUBMODULAR FUNCTIONS C. J. H. McDIARMID 1. Preliminaries We prove that a submodular function (not necessarily non-decreasing) induces in a natural way a pre-independence structure, thus extending known results [1, 3]. For any set A, the collection of subsets of A is denoted by 2 , and the cardinal of A is denoted by \A\. All functions, including |. |, are assumed to take values from {0, 1, ..., oo}. For sets A, B we write A c c B if A is a finite subse t of B. Given a set S, a non-empty set $ £ 2 is a pre-independence structure on S if it satisfies (1) ifXeS and 7 £ X, then 7e^ ; (2) if X, Y e<£ are finite and | Y\ = \X\ +1, then there exists y e Y\X such that X\j{y}<=£. We say that & is of finite character if it satisfies (3) if X c S and Y e<£ for every YccI , then Ze<?. An independence structure is a pre-independence structure of finite character. The rank function p of a pre-independence structure <f on S is defined by pX = sup{|Y|: YeS, YcczX} (XgS). Given a

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Mar 1, 1973

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