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A first efficient algorithm for enumerating all the extreme points of a bisubmodular polyhedron

Yasuko Matsui, Takeshi Naitoh, Ping Zhan

TL;DR

This paper presents an algorithm that enumerates all the extreme points of a bisubmodular polyhedron in $\mathcal{O}(n^4|V|)$ time and space complexity, where n is the dimension of underlying space and V is the set of outputs.

Abstract

Efficiently enumerating all the extreme points of a polytope identified by a system of linear inequalities is a well-known challenge issue.We consider a special case and present an algorithm that enumerates all the extreme points of a bisubmodular polyhedron in $\mathcal{O}(n^4|V|)$ time and $\mathcal{O}(n^2)$ space complexity, where $ n$ is the dimension of underlying space and $V$ is the set of outputs. We use the reverse search and signed poset linked to extreme points to avoid the redundant search. Our algorithm is a generalization of enumerating all the extreme points of a base polyhedron which comprises some combinatorial enumeration problems.

A first efficient algorithm for enumerating all the extreme points of a bisubmodular polyhedron

TL;DR

This paper presents an algorithm that enumerates all the extreme points of a bisubmodular polyhedron in time and space complexity, where n is the dimension of underlying space and V is the set of outputs.

Abstract

Efficiently enumerating all the extreme points of a polytope identified by a system of linear inequalities is a well-known challenge issue.We consider a special case and present an algorithm that enumerates all the extreme points of a bisubmodular polyhedron in time and space complexity, where is the dimension of underlying space and is the set of outputs. We use the reverse search and signed poset linked to extreme points to avoid the redundant search. Our algorithm is a generalization of enumerating all the extreme points of a base polyhedron which comprises some combinatorial enumeration problems.
Paper Structure (8 sections, 7 theorems, 39 equations, 3 figures, 1 table)

This paper contains 8 sections, 7 theorems, 39 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Bisubmodular polyhedra and the extreme point theorem
  4. Signed posets and the adjacency of extreme points
  5. Reverse search for enumeration
  6. A polynomial enumeration algorithm
  7. Concluding remarks
  8. The proof of Lemma \ref{['capacity']}

Key Result

Theorem 2.1

A vector $x\in \mathbb{R}^N$ is an extreme point of ${\rm P}_*(f)$ if and only if there exists a chain such that for each $i=1,2, \ldots ,n$, where $(U_0, W_0)=(\emptyset,\emptyset)$.

Figures (3)

  • Figure 1: An example of the strict bisubmodular function. (The flow of the computations: $x_i \to {\rm sat}(x_i) \to {\rm dep}(x_i) \to (G(x_i)) \ {\cal H}(x_i ) \to {\cal I}(x_i) \to \tilde{G}(x_i)$, $i=1,2$).
  • Figure 2: An example of the non-strict bisubmodular function.
  • Figure 3: An example of the execution of Algorithm 2.

Theorems & Definitions (7)

  • Theorem 2.1: Extreme point theorem cubcck
  • Proposition 2.2
  • Theorem 2.3: af4
  • Proposition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Proposition 3.4