Signed ring families and signed posets

The one-to-one correspondence between finite distributive lattices and finite partially ordered sets (posets) is a well-known theorem of G. Birkhoff. This implies a nice representation of any distributive lattice by its corresponding poset, where the size of the former (distributive lattice) is often exponential in the size of the underlying set of the latter (poset). A lot of engineering and economic applications bring us distributive lattices as a ring family of sets which is closed with respect to the set union and intersection. When it comes to a ring family of sets, the underlying set is partitioned into subsets (or components) and we have a poset structure on the partition. This is a set-theoretical variant of the Birkhoff theorem revealing the correspondence between finite ring families and finite posets on partitions of the underlying sets, which was pursued by Masao Iri around 1978, especially concerned with what is called the principal partition of discrete systems such as graphs, matroids, and polymatroids. In the present paper we investigate a signed-set version of the Birkhoff-Iri decomposition in terms of signed ring family, which corresponds to Reiner's result on signed posets, a signed counterpart of the Birkhoff theorem. We show that given a signed ring family, we have a signed partition of the underlying set together with a signed poset on the signed partition which represents the given signed ring family. This representation is unique up to certain reflections

A Min-Max . . . Functions and Its Implications

A. Huber and V. Kolmogorov (ISCO 2012) introduced a concept of k-submodular function as a generalization of ordinary submodular (set) functions and bisubmodular functions and obtained a min-max theorem for minimization of k-submodular functions. Also F. Kuivinen (2011) considered submodular functions on (product lattices of) diamonds and showed a min-max theorem for minimization of submodular functions on diamonds. In the present paper we consider a common generalization of k-submodular functions and submodular functions on diamonds, which we call a transversal submodular function (or a t-submodular function, for short). We show a min-max theorem for minimization of t-submodular functions in terms of a new norm composed of ℓ1 and ℓ ∞ norms. This reveals a relationship between the obtained min-max theorem and that for minimization of ordinary submodular set functions due to J. Edmonds (1970). We also show how our min-max theorem for t-submodular functions can be used to prove the min-max theorem for k-submodular functions by Huber and Kolmogorov and that for submodular functions on diamonds by Kuivinen. Moreover, we show a counterexample to a characterization, given by Huber and Kolmogorov (ISCO 2012), of extreme points of the k-submodular polyhedron and make it a correct one by fixing a flaw therein

A min-max theorem for bisubmodular polyhedra

SIGLEAvailable from TIB Hannover: RN 4052(94818) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

A min-max theorem for bisubmodular polyhedra

SIGLEAvailable from TIB Hannover: RN 4052(94818) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman