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  • 1
    Electronic Resource
    Electronic Resource
    A recipe for semidefinite relaxation for (0,1)-quadratic programming (1995)
    Springer
    Journal of global optimization 7 (1995), S. 51-73 
    Details
    ISSN: 1573-2916
    Keywords: Quadratic boolean programming ; semidefinite programming ; bounds ; Lagrangian duality ; parametric programming ; trust region subproblems ; minmax eigenvalue problems ; quadratic assignment problem ; graph partitioning ; max-clique ; theta function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We review various relaxations of (0,1)-quadratic programming problems. These include semidefinite programs, parametric trust region problems and concave quadratic maximization. All relaxations that we consider lead to efficiently solvable problems. The main contributions of the paper are the following. Using Lagrangian duality, we prove equivalence of the relaxations in a unified and simple way. Some of these equivalences have been known previously, but our approach leads to short and transparent proofs. Moreover we extend the approach to the case of equality constrained problems by taking the squared linear constraints into the objective function. We show how this technique can be applied to the Quadratic Assignment Problem, the Graph Partition Problem and the Max-Clique Problem. Finally we show our relaxation to be best possible among all quadratic majorants with zero trace.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01100205
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  • 2
    Electronic Resource
    Electronic Resource
    Quadratic assignment problems on series-parallel digraphs (1986)
    Springer
    Mathematical methods of operations research 30 (1986), S. A161 
    Details
    ISSN: 1432-5217
    Keywords: Quadratic assignment problems ; series-parallel digraphs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Christofides hat gezeigt, daß das quadratische Zuordnungsproblem (QZP) auf isomorphen Bäumen polynomial lösbar ist. Der Ansatz von Christofides wird auf minimale serienparallele Digraphen verallgemeinert, und es wird gezeigt, daß das resultierende QZP äquivalent zum allgemeinen QZP ist, also NP-schwer. Die Einschränkung des Problems auf diejenigen serien-parallelen Digraphen, die keine bipartiten Teilgraphen enthalten, ist wieder polynomial lösbar. Dafür wird ein Algorithmus angegeben.
    Notes: Abstract Christofides has shown that the quadratic assignment problem (QAP) on isomorphic trees is solvable in polynomial time by dynamic programming. We generalize the approach to minimal series-parallel digraphs and show that this problem is equivalent to the general QAP, thus NP-hard. However the restriction to the subclass of series-parallel digraphs not containing bipartite subgraphs again is solvable in polynomial time. An algorithm for this class is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01919179
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    Paper (German National Licenses)
    Fulltext
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