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

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

2008-01-24

arxiv.org/abs/0801.3790v2

Computer Science

Computational Complexity

Title typo fixed

Scientific paper

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)].

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