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  • 1
    Electronic Resource
    Electronic Resource
    On the symmetric travelling salesman problem I: Inequalities (1979)
    Springer
    Mathematical programming 16 (1979), S. 265-280 
    Details
    ISSN: 1436-4646
    Keywords: Linear Inequalities ; Convex Polytopes ; Facets ; Travelling Salesman Problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We investigate several classes of inequalities for the symmetric travelling salesman problem with respect to their facet-defining properties for the associated polytope. A new class of inequalities called comb inequalities is derived and their number shown to grow much faster with the number of cities than the exponentially growing number of subtour-elimination constraints. The dimension of the travelling salesman polytope is calculated and several inequalities are shown to define facets of the polytope. In part II (“On the travelling salesman problem II: Lifting theorems and facets”) we prove that all subtour-elimination and all comb inequalities define facets of the symmetric travelling salesman polytope.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582116
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    On the acyclic subgraph polytope (1985)
    Springer
    Mathematical programming 33 (1985), S. 28-42 
    Details
    ISSN: 1436-4646
    Keywords: Acyclic Subgraph Problem ; Feedback Arc Set Problem ; Facets of Polyhedra ; Polynomial Time Algorithms ; Weakly Acyclic Digraphs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract 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 in polynomial time for weakly acyclic digraphs. This generalizes a result of Lucchesi for planar digraphs.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582009
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Facets of the linear ordering polytope (1985)
    Springer
    Mathematical programming 33 (1985), S. 43-60 
    Details
    ISSN: 1436-4646
    Keywords: Facets of Polyhedra ; Linear Ordering Problem ; Triangulation Problem ; Permutation Problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract LetD n be the complete digraph onn nodes, and letP LO n denote the convex hull of all incidence vectors of arc sets of linear orderings of the nodes ofD n (i.e. these are exactly the acyclic tournaments ofD n ). We show that various classes of inequalities define facets ofP LO n , e.g. the 3-dicycle inequalities, the simplek-fence inequalities and various Möbius ladder inequalities, and we discuss the use of these inequalities in cutting plane approaches to the triangulation problem of input-output matrices, i.e. the solution of permutation resp. linear ordering problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582010
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    Paper (German National Licenses)
    Fulltext
  • 4
    Electronic Resource
    Electronic Resource
    On the symmetric travelling salesman problem II: Lifting theorems and facets (1979)
    Springer
    Mathematical programming 16 (1979), S. 281-302 
    Details
    ISSN: 1436-4646
    Keywords: Linear Inequalities ; Convex Polytopes ; Facets ; Lifting Theorems ; Travelling Salesman Problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Four lifting theorems are derived for the symmetric travelling salesman polytope. They provide constructions and state conditions under which a linear inequality which defines a facet of then-city travelling salesman polytope retains its facetial property for the (n + m)-city travelling salesman polytope, wherem ≥ 1 is an arbitrary integer. In particular, they permit a proof that all subtour-elimination as well as comb inequalities define facets of the convex hull of tours of then-city travelling salesman problem, wheren is an arbitrary integer.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582117
    Library Location Call Number Volume/Issue/Year Availability
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    Paper (German National Licenses)
    Fulltext
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