6,014 research outputs found
The QAP-Polytope and the Star-Transformation
Polyhedral Combinatorics has been successfully applied to obtain considerable algorithmic progress towards the solution of many prominent hard combinatorial optimization problems. Until very recently, the quadratic assignment problem (QAP) was one of the few exceptions. Recent work of Rijal (1995) and Padberg and Rijal (1996) has on the one hand yielded some basic facts about the associated quadratic assignment polytope, but has on the other hand shown that investigations even of the very basic questions (like the dimension, the affine hull, and the trivial facets) soon become extremely complicated. In this paper, we propose an isomorphic transformation of the ''natural'' realization of the quadratic assignment polytope, which simplifies the polyhedral investigations enormously. We demonstrate this by giving short proofs of the basic results on the polytope that indicate that, exploiting the techniques developed in this paper, deeper polyhedral investigations of the QAP now become possible. Moreover, an 'ìnductive construction'' of the QAP-Polytope is derived that might be useful in branch-and-cut algorithms
Acyclic orientations with path constraints
Many well-known combinatorial optimization problems can be stated over the
set of acyclic orientations of an undirected graph. For example, acyclic
orientations with certain diameter constraints are closely related to the
optimal solutions of the vertex coloring and frequency assignment problems. In
this paper we introduce a linear programming formulation of acyclic
orientations with path constraints, and discuss its use in the solution of the
vertex coloring problem and some versions of the frequency assignment problem.
A study of the polytope associated with the formulation is presented, including
proofs of which constraints of the formulation are facet-defining and the
introduction of new classes of valid inequalities
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
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