Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Note on Ordinal Submodularity

Satoru Fujishige, Ryuhei Mizutani

Abstract

Notions of ordinal submodularity/supermodularity have been introduced and studied in the literature. We consider several classes of ordinally submodular functions defined on finite Boolean lattices and give characterizations of the set of minimizers of ordinally submodular functions.

A Note on Ordinal Submodularity

Abstract

Notions of ordinal submodularity/supermodularity have been introduced and studied in the literature. We consider several classes of ordinally submodular functions defined on finite Boolean lattices and give characterizations of the set of minimizers of ordinally submodular functions.
Paper Structure (6 sections, 4 theorems, 7 equations, 1 figure)

This paper contains 6 sections, 4 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ordinal Notions of Submodularity
  3. Minimizers of Ordinally Submodular Functions
  4. (Q1)-submodular functions
  5. (Q4)-submodular functions
  6. Discussions

Key Result

Lemma 2.1

Any (Q3)-submodular function is (Q4)-submodular. (Proof) Let $f: 2^E\to(\mathbb{P},\le)$ be any (Q3)-submodular function and consider arbitrary $X,Y\in 2^E$. If $f(X)\ge f(X\cap Y)$, then we have Also, if $f(X)< f(X\cap Y)$, from (Q3) we have $f(X\cup Y)\le f(Y)$. Hence we have It follows that the max-min relation of (Q4) holds for all $X, Y\in 2^E$. $\Box$

Figures (1)

  • Figure 1: Relationship among (Qi) for $i=1,2,3,4$.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Lemma 2.1
  • Remark 3
  • Lemma 3.1
  • Theorem 3.2
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 3.3