Cones of Elementary Imsets and Supermodular Functions: A Review and Some New Results

Abstract

In this paper we give a review of the method of imsets introduced by Studený 1 from a geometric point of view. Elementary imsets span a polyhedral cone and its dual cone is the cone of supermodular functions. We review basic facts on the structure of these cones. Then we derive some new results on the following topics: i) extreme rays of the cone of standardized supermodular functions, ii) faces of the cones, iii) small relations among elementary imsets, and iv) some computational results on Markov basis for the toric ideal defined by elementary imsets.

Core-based criterion for extreme supermodular functions

“…By (15), one also has S ∈ S m π = S π . Since the conditions (a)-(b) define a linear space in R Υ×N and both x m and y satisfy them, for any ε ∈ R, the vector z ε must satisfy them as well.…”

“…Further considerations are done with a fixed set S ⊆ N. One can certainly find and fix π ∈ Υ with S ∈ C π . By (15), one also has S ∈ S m π = S π . Since the conditions (a)-(b) define a linear space in R Υ×N and both x m and y satisfy them, for any ε ∈ R, the vector z ε must satisfy them as well.…”

“…To verify (19) realize that, for τ, π ∈ Υ and S ∈ C τ ∩ C π , (15) gives C τ ⊆ S m τ = S τ and C π ⊆ S m π = S π and then the condition (b) for y implies i∈S y(τ, i) = i∈S y(π, i), which was desired.…”

“…Extreme supermodular functions also establish quite an important class of inequalities that are used in (integer) linear programming approach to learning Bayesian network structure; see [36, § 3.1] or [5, § 7]. There were attempts to classify extreme supermodular functions and the operations with them [35,15]; see also the open problems from § 9.1.2 of [34]. 3.…”

“…In connection with the application of extreme supermodular functions in testing conditional independence implication by means of a computer a catalogue of all (types of) extreme supermodular functions over 5 variables has been created [6]. The paper [2] contains several interesting results on (linear) operations with supermodular functions which preserve extremality. The aim of this technical report is to gather the results on operations with set functions preserving supermodularity and put the observations from [2] in this context.…”