Characterization of the Vertices and Extreme Directions of the Negative Cycles Polyhedron and Hardness of Generating Vertices of 0/1-Polyhedra

TL;DR

This paper fully characterizes the vertices and extreme directions of the negative cycles polyhedron and proves that generating all vertices of 0/1-polyhedra is computationally hard unless P=NP.

Contribution

It provides a complete characterization of the vertices and extreme directions of the negative cycles polyhedron and establishes the hardness of generating all vertices of 0/1-polyhedra.

Findings

Complete characterization of vertices and extreme directions of the negative cycles polyhedron.

Proves no output polynomial-time algorithm exists for generating all vertices of 0/1-polyhedra unless P=NP.

Contrasts the hardness result with polynomial vertex enumeration for 0/1-polytopes.

Abstract

Given a graph $G=(V,E)$ and a weight function on the edges $w:E\mapsto\RR$, we consider the polyhedron $P(G,w)$ of negative-weight flows on $G$, and get a complete characterization of the vertices and extreme directions of $P(G,w)$. As a corollary, we show that, unless $P=NP$, there is no output polynomial-time algorithm to generate all the vertices of a 0/1-polyhedron. This strengthens the NP-hardness result of Khachiyan et al. (2006) for non 0/1-polyhedra, and comes in contrast with the polynomiality of vertex enumeration for 0/1-polytopes \cite{BL98} [Bussieck and L\"ubbecke (1998)].