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    <id>942</id>
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    <language>eng</language>
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    <pageLast/>
    <pageNumber/>
    <edition/>
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    <type>reportzib</type>
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    <completedDate>2006-11-17</completedDate>
    <publishedDate>2006-11-17</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Orbitopal Fixing</title>
    <abstract language="eng">The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branch-and-cut algorithms if the permutation of the subsets of the partition is irrelevant. This kind of symmetry unnecessarily blows up the branch-and-cut tree. We present a general tool, called orbitopal fixing, for enhancing the capabilities of branch-and-cut algorithms in solving this kind of symmetric integer programming models. We devise a linear time algorithm that, applied at each node of the branch-and-cut tree, removes redundant parts of the tree produced by the above mentioned permutations. The method relies on certain polyhedra, called orbitopes, which have been investigated in (Kaibel and Pfetsch (2006)). However, it does not add inequalities to the model, and thus, it does not increase the difficulty of solving the linear programming relaxations. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem motivated from frequency planning in mobile telecommunication networks.</abstract>
    <identifier type="serial">06-48</identifier>
    <identifier type="opus3-id">942</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-9422</identifier>
    <enrichment key="SourceTitle">Appeared in: Proc. of the 12th Integer Programming and Combinatorial Optimization Conference (IPCO) M. Fischetti and D. Williamson (eds.), LNCS 4513, Springer-Verlag, 74-88</enrichment>
    <author>Volker Kaibel</author>
    <author>Matthias Peinhardt</author>
    <author>Marc Pfetsch</author>
    <series>
      <title>ZIB-Report</title>
      <number>06-48</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>symmetry breaking</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>variable fixing</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>orbitopes</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="52B12">Special polytopes (linear programming, centrally symmetric, etc.)</collection>
    <collection role="msc" number="90C10">Integer programming</collection>
    <collection role="msc" number="90C57">Polyhedral combinatorics, branch-and-bound, branch-and-cut</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="projects" number="MATHEON-B12:IPSym">MATHEON-B12:IPSym</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/942/ZR_06_48.pdf</file>
    <file>https://opus4.kobv.de/opus4-zib/files/942/ZR-06-48.pdf</file>
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    <file>https://opus4.kobv.de/opus4-zib/files/942/ZR-06-48.ps</file>
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  <doc>
    <id>910</id>
    <completedYear/>
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    <language>eng</language>
    <pageFirst/>
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    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2006-03-28</completedDate>
    <publishedDate>2006-03-28</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Packing and Partitioning Orbitopes</title>
    <abstract language="eng">We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which arise from restrictions to matrices with at most or exactly one 1-entry in each row, respectively. The goal of investigating these polytopes is to gain insight into ways of breaking certain symmetries in integer programs by adding constraints, e.g., for a well-known formulation of the graph coloring problem. We provide a thorough polyhedral investigation of packing and partitioning orbitopes for the cases in which the group acting on the columns is the cyclic group or the symmetric group. Our main results are complete linear inequality descriptions of these polytopes by facet-defining inequalities. For the cyclic group case, the descriptions turn out to be totally unimodular, while for the symmetric group case, both the description and the proof are more involved. The associated separation problems can be solved in linear time.</abstract>
    <identifier type="serial">06-17</identifier>
    <identifier type="opus3-id">911</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-9104</identifier>
    <enrichment key="SourceTitle">Appeared in: Mathematical Programming 114 (2008) 1-36</enrichment>
    <author>Volker Kaibel</author>
    <author>Marc Pfetsch</author>
    <series>
      <title>ZIB-Report</title>
      <number>06-17</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>integer programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>symmetry breaking</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>lexicographic representatives</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="52B12">Special polytopes (linear programming, centrally symmetric, etc.)</collection>
    <collection role="msc" number="90C10">Integer programming</collection>
    <collection role="msc" number="90C57">Polyhedral combinatorics, branch-and-bound, branch-and-cut</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="projects" number="MATHEON-B12:IPSym">MATHEON-B12:IPSym</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/910/ZR-06-17.pdf</file>
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