M. Studeny:
Basic facts concerning extreme supermodular functions. Research report n. 2359,
Institute of Information Theory and Automation, Prague, December 2016.
- Abstract
-
Elementary facts and observations on the cone of supermodular set functions are recalled.
The manuscript deals with such operations with set functions which preserve supermodularity
and the emphasis is put on those such operations which even preserve extremity
(of a supermodular function). These involve a few self-transformations of the cone
of supermodular set functions. Moreover, projections to the (less-dimensional) linear space of
set functions for a subset of the variable set are discussed. Finally, several extensions to the
(more-dimensional) linear space of set functions for a superset of the variable set are shown
to be both preserving supermodularity and extremality.
- AMS classification 68T30
- Keywords
- supermodular function
- standardizations
- extreme supermodular function
- permutational transformation
- reflection
- lifting
- support
- minor
- coarsening
- contraction
- modular extension
- replication
- A
pdf version (434kB) is available. See also its
arxiv version 1612.06599
The report builds on these publications:
- T. Kashimura, T. Sei, A. Takemura, K. Tanaka:
Cones of elementary imsets and supermodular functions: a review and some new results.
In Proceedings of the 2nd CREST-SBM International
Conference Harmony of Groebner Bases and the Modern Industrial
Society, World Scientific 2012, pp. 117-152.
- F. Bach:
Convex analysis and optimization with submodular functions: a tutorial.
A manuscript from November 2010, available at
arXiv:1010.4207v2
- M. Studeny, R.R. Bouckaert, T. Kocka:
Extreme supermodular set functions over five variables.
Research report n. 1977,
Institute of Information Theory and Automation,
Prague, January 2000. (32 pages)