On the acyclic subgraph polytope

@article{Grtschel1985OnTA,
  title={On the acyclic subgraph polytope},
  author={Martin Gr{\"o}tschel and Michael J{\"u}nger and Gerhard Reinelt},
  journal={Mathematical Programming},
  year={1985},
  volume={33},
  pages={28-42}
}
The acyclic subgraph problem can be formulated as follows. Given a digraph with arc weights, find a set of arcs containing no directed cycle and having maximum total weight. We investigate this problem from a polyhedral point of view and determine several classes of facets for the associated acyclic subgraph polytope. We also show that the separation problem for the facet defining dicycle inequalities can be solved in polynomial time. This implies that the acyclic subgraph problem can be solved… 

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References

SHOWING 1-8 OF 8 REFERENCES

The ellipsoid method and its consequences in combinatorial optimization

TLDR
The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.

A Cutting Plane Algorithm for the Linear Ordering Problem

TLDR
A new algorithm is reported on that could triangulate all input-output matrices, of size up to 60 × 60, available to us within acceptable time bounds and compare favorably with the results of existing codes.

A Minimax Theorem for Directed Graphs

This minimax equality was conjectured about a decade ago by one of the authors ([7; page 43], [8], [9]) and, independently, by Neil Robertson. It arose in the study of a problem posed several years

Computers and Intractability: A Guide to the Theory of NP-Completeness

TLDR
The experiences, understandings, and beliefs that guide the professional practices of teacher educators are explored, and the book paints a picture of a profession that offers huge rewards, alongside challenges and frustrations.

Optimal triangulation of large real world input-output matrices

In this paper we present optimum triangulations of a large number of input-output matrices. In particular, we report about a series of (44,44)-matrices of the years 1959, 1965, 1970, 1975 of the

Facets of the linear ordering polytope

TLDR
It is shown that various classes of inequalities define facets ofPLOn, e.g. the 3-dicycle inequalities, the simplek-fence inequalities and various Möbius ladder inequalities, and the use of these inequalities in cutting plane approaches to the triangulation problem of input-output matrices is discussed.