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

TL;DR

This paper introduces an efficient algorithm for enumerating all extreme points of a bisubmodular polyhedron, significantly improving computational complexity for this specific class of polytopes.

Contribution

The paper presents the first efficient algorithm for enumerating all extreme points of a bisubmodular polyhedron, utilizing reverse search and signed posets to optimize performance.

Findings

Algorithm runs in $ ext{O}(n^4|V|)$ time and $ ext{O}(n^2)$ space.

Generalizes enumeration methods from base polyhedra to bisubmodular polyhedra.

Reduces redundant search through novel use of signed posets.

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.