Polynomial time vertex enumeration of convex polytopes of bounded branch-width

TL;DR

This paper introduces a polynomial time algorithm for vertex enumeration of convex polytopes with bounded branch-width, advancing understanding of polyhedral enumeration in high-dimensional spaces.

Contribution

It applies branch-decomposition to vertex enumeration, introducing the concept of k-module and providing a total polynomial time algorithm for polytopes with bounded branch-width.

Findings

Algorithm runs in polynomial time for bounded branch-width cases

Introduces the concept of k-module related to linear matroid separators

Establishes a connection between branch-width and efficient vertex enumeration

Abstract

Over the last years the vertex enumeration problem of polyhedra has seen a revival in the study of metabolic networks, which increased the demand for efficient vertex enumeration algorithms for high-dimensional polyhedra given by inequalities. It is a famous and long standing open question in polyhedral theory and computational geometry whether the vertices of a polytope (bounded polyhedron), described by a set of linear constraints, can be enumerated in total polynomial time. In this paper we apply the concept of branch-decomposition to the vertex enumeration problem of polyhedra $P = \{x : Ax = b, x \geq 0\}$. For this purpose, we introduce the concept of $k$-module and show how it relates to the separators of the linear matroid generated by the columns of $A$. We then use this to present a total polynomial time algorithm for polytopes $P$ for which the branch-width of the linear matroid generated by $A$ is bounded by a constant $k$.