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    <id>197</id>
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    <language>eng</language>
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    <completedDate>1995-11-29</completedDate>
    <publishedDate>1995-11-29</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Optimum Path Packing on Wheels: The Consecutive Case</title>
    <abstract language="eng">We show that, given a wheel with nonnegative edge lengths and pairs of terminals located on the wheel's outer cycle such that the terminal pairs are in consecutive order, then a path packing, i.~e., a collection of edge disjoint paths connecting the given terminal pairs, of minimum length can be found in strongly polynomial time. Moreover, we exhibit for this case a system of linear inequalities that provides a complete and nonredundant description of the path packing polytope, which is the convex hull of all incidence vectors of path packings and their supersets.</abstract>
    <identifier type="serial">SC-95-31</identifier>
    <identifier type="opus3-id">197</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-1976</identifier>
    <enrichment key="SourceTitle">Appeared in: Computers Math. Applic., Vol. 31, No. 11, pp. 23-35, 1996</enrichment>
    <author>Martin Grötschel</author>
    <author>Alexander Martin</author>
    <author>Robert Weismantel</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-95-31</number>
    </series>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="groetschel">Grötschel, Martin</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/197/SC-95-31.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/197/SC-95-31.pdf</file>
  </doc>
  <doc>
    <id>132</id>
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    <language>eng</language>
    <pageFirst/>
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    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1994-01-20</completedDate>
    <publishedDate>1994-01-20</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">The Steiner Tree Packing Problem in VLSI-Design.</title>
    <abstract language="eng">In this paper we describe several versions of the routing problem arising in VLSI design and indicate how the Steiner tree packing problem can be used to model these problems mathematically. We focus on switchbox routing problems and provide integer programming formulations for routing in the knock-knee and in the Manhattan model. We give a brief sketch of cutting plane algorithms that we developed and implemented for these two models. We report on computational experiments using standard test instances. Our codes are able to determine optimum solutions in most cases, and in particular, we can show that some of the instances have no feasible solution if Manhattan routing is used instead of knock-knee routing.</abstract>
    <identifier type="serial">SC-94-02</identifier>
    <identifier type="opus3-id">132</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-1328</identifier>
    <enrichment key="SourceTitle">Appeared in: Mathematical Programming 78 (1997) 265-281</enrichment>
    <author>Martin Grötschel</author>
    <author>Alexander Martin</author>
    <author>Robert Weismantel</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-94-02</number>
    </series>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="groetschel">Grötschel, Martin</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/132/SC-94-02.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/132/SC-94-02.pdf</file>
  </doc>
  <doc>
    <id>122</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
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    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1993-10-27</completedDate>
    <publishedDate>1993-10-27</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Optimum Path Packing on Wheels: The Noncrossing Case.</title>
    <abstract language="eng">We show that, given a wheel with nonnegative edge lengths and pairs of terminals located on the wheel's outer cycle such that no two terminal pairs cross, then a path packing, i.~e.,a collection of edge disjoint paths connecting the given terminal pairs, of minimum length can be found in strongly polynomial time. Moreover, we exhibit for this case a system of linear inequalities that provides a complete and nonredundant description of the path packing polytope, which is the convex hull of all incidence vectors of path packings and their supersets.</abstract>
    <identifier type="serial">SC-93-26</identifier>
    <identifier type="opus3-id">121</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-1225</identifier>
    <enrichment key="SourceTitle">See SC 95-31 for an updated version</enrichment>
    <author>Martin Grötschel</author>
    <author>Alexander Martin</author>
    <author>Robert Weismantel</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-93-26</number>
    </series>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="groetschel">Grötschel, Martin</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/122/SC-93-26.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/122/SC-93-26.pdf</file>
  </doc>
  <doc>
    <id>78</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1992-04-09</completedDate>
    <publishedDate>1992-04-09</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Packing Steiner Trees: Polyhedral Investigations.</title>
    <abstract language="eng">Let $G=(V,E)$ be a graph and $T\subseteq V$ be a node set. We call an edge set $S$ a Steiner tree with respect to $T$ if $S$ connects all pairs of nodes in $T$. In this paper we address the following problem, which we call the weighted Steiner tree packing problem. Given a graph $G=(V,E)$ with edge weights $w_e$, edge capacities $c_e, e \in E,$ and node sets $T_1,\ldots,T_N$, find edge sets $S_1,\ldots,S_N$ such that each $S_k$ is a Steiner tree with respect to $T_k$, at most $c_e$ of these edge sets use edge $e$ for each $e\in E$, and such that the sum of the weights of the edge sets is minimal. Our motivation for studying this problem arises from the routing problem in VLSI-design, where given sets of points have to be connected by wires. We consider the Steiner tree packing Problem from a polyhedral point of view and define an appropriate polyhedron, called the Steiner tree packing polyhedron. The goal of this paper is to (partially) describe this polyhedron by means of inequalities. It turns out that, under mild assumptions, each inequality that defines a facet for the (single) Steiner tree polyhedron can be lifted to a facet-defining inequality for the Steiner tree packing polyhedron. The main emphasis of this paper lies on the presentation of so-called joint inequalities that are valid and facet-defining for this polyhedron. Inequalities of this kind involve at least two Steiner trees. The classes of inequalities we have found form the basis of a branch &amp; cut algorithm. This algorithm is described in our companion paper SC 92-09.</abstract>
    <identifier type="serial">SC-92-08</identifier>
    <identifier type="opus3-id">77</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-786</identifier>
    <identifier type="doi">10.1007/BF02592085</identifier>
    <enrichment key="SourceTitle">Appeared in: Mathematical Programming 72 (1996) 101-123</enrichment>
    <author>Martin Grötschel</author>
    <author>Alexander Martin</author>
    <author>Robert Weismantel</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-92-08</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>cutting planes</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Facets</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Packing</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>polyhedron</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Steiner tree</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="groetschel">Grötschel, Martin</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/78/SC-92-08.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/78/SC-92-08.pdf</file>
  </doc>
  <doc>
    <id>79</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1992-04-09</completedDate>
    <publishedDate>1992-04-09</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Packing Steiners Trees: A Cutting Plane Algorithm and Computational Results.</title>
    <abstract language="eng">In this paper we describe a cutting plane algorithm for the Steiner tree packing problem. We use our algorithm to solve some switchbox routing problems of VLSI-design and report on our computational experience. This includes a brief discussion of separation algorithms, a new LP-based primal heuristic and implementation details. The paper is based on the polyhedral theory for the Steiner tree packing polyhedron developed in our companion paper SC 92-8 and meant to turn this theory into an algorithmic tool for the solution of practical problems.</abstract>
    <identifier type="serial">SC-92-09</identifier>
    <identifier type="opus3-id">78</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-792</identifier>
    <enrichment key="SourceTitle">Appeared in: Mathematical Programming 72 (1996) 125-145</enrichment>
    <author>Martin Grötschel</author>
    <author>Alexander Martin</author>
    <author>Robert Weismantel</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-92-09</number>
    </series>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="groetschel">Grötschel, Martin</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/79/SC-92-09.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/79/SC-92-09.pdf</file>
  </doc>
  <doc>
    <id>95</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
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    <type>reportzib</type>
    <publisherName/>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1993-01-13</completedDate>
    <publishedDate>1993-01-13</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Some Integer Programs Arising in the Design of Main Frame Computers.</title>
    <abstract language="eng">In this paper we describe and discuss a problem that arises in the (global) design of a main frame computer. The task is to assign certain functional units to a given number of so called multi chip modules or printed circuit boards taking into account many technical constraints and minimizing a complex objective function. We describe the real world problem. A thorough mathematical modelling of all aspects of this problem results in a rather complicated integer program that seems to be hopelessly difficult -- at least for the present state of integer programming technology. We introduce several relaxations of the general model, which are also $NP$-hard, but seem to be more easily accessible. The mathematical relations between the relaxations and the exact formulation of the problem are discussed as well.</abstract>
    <identifier type="serial">SC-92-25</identifier>
    <identifier type="opus3-id">94</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-957</identifier>
    <enrichment key="SourceTitle">Appeared in: Zeitschrift für Operations Research 38 1 (1993) pp. 77-100.</enrichment>
    <author>Carlos E. Ferreira</author>
    <author>Martin Grötschel</author>
    <author>Stefan Kiefl</author>
    <author>Ludwig Krispenz</author>
    <author>Alexander Martin</author>
    <author>Robert Weismantel</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-92-25</number>
    </series>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="groetschel">Grötschel, Martin</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/95/SC-92-25.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/95/SC-92-25.pdf</file>
  </doc>
  <doc>
    <id>96</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1992-12-02</completedDate>
    <publishedDate>1992-12-02</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Routing in Grid Graphs by Cutting Planes.</title>
    <abstract language="eng">{\def\N{{\cal N}} \def\R{\hbox{\rm I\kern-2pt R}} \def\MN{{\rm I\kern-2pt N}} In this paper we study the following problem, which we call the weighted routing problem. Let be given a graph $G=(V,E)$ with non-negative edge weights $w_e\in\R_+$ and integer edge capacities $c_e\in\MN$ and let $\N=\{T_1,\ldots,T_N\}$, $N\ge 1$, be a list of node sets. The weighted routing problem consists in finding edge sets $S_1,\ldots,S_N$ such that, for each $k\in\{1,\ldots,N\}$, the subgraph $(V(S_k),S_k)$ contains an $[s,t]$-path for all $s,t\in T_k$, at most $c_e$ of these edge sets use edge $e$ for each $e\in E$, and such that the sum of the weights of the edge sets is minimal. Our motivation for studying this problem arises from the routing problem in VLSI-design, where given sets of points have to be connected by wires. We consider the weighted routing problem from a polyhedral point of view. We define an appropriate polyhedron and try to (partially) describe this polyhedron by means of inequalities. We briefly sketch our separation algorithms for some of the presented classes of inequalities. Based on these separation routines we have implemented a branch and cut algorithm. Our algorithm is applicable to an important subclass of routing problems arising in VLSI-design, namely to problems where the underlying graph is a grid graph and the list of node sets is located on the outer face of the grid. We report on our computational experience with this class of problem instances.}</abstract>
    <identifier type="serial">SC-92-26</identifier>
    <identifier type="opus3-id">95</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-967</identifier>
    <enrichment key="SourceTitle">Appeared in: Integer Programming and Combinatorial Optimization - IPCO-03. Proc. of a Conf., Erice, April 29 - May 1, 1993. Core, Louvain-la-Neuve, Belgium, 1993, pp. 447-461. An extended version appeared in: ZOR - Mathematical Methods of Operations Research 41 (1995) pp. 255-275</enrichment>
    <author>Martin Grötschel</author>
    <author>Alexander Martin</author>
    <author>Robert Weismantel</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-92-26</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>routing in VLSI-design</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Steiner tree</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Steiner tree packing</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>cutting plane algorithm</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="groetschel">Grötschel, Martin</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/96/SC-92-26.pdf</file>
  </doc>
  <doc>
    <id>97</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
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    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1992-12-02</completedDate>
    <publishedDate>1992-12-02</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Packing Steiner Trees: Further Facets.</title>
    <abstract language="eng">In this paper we continue the investigations in [GMW92a] for the \def\sbppo{Steiner tree packing polyhedron} \sbppo. We present several new classes of valid inequalities and give sufficient (and necessary) conditions for these inequalities to be facet-defining. It is intended to incorporate these inequalities into an existing cutting plane algorithm that is applicable to practical problems arising in the design of electronic circuits.</abstract>
    <identifier type="serial">SC-93-01</identifier>
    <identifier type="opus3-id">96</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-975</identifier>
    <identifier type="doi">10.1006/eujc.1996.0004</identifier>
    <enrichment key="SourceTitle">Appeared in: Europ. J. Combinatorics (1996) 17, 39-52</enrichment>
    <author>Martin Grötschel</author>
    <author>Alexander Martin</author>
    <author>Robert Weismantel</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-93-01</number>
    </series>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="groetschel">Grötschel, Martin</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/97/SC-93-01.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/97/SC-93-01.pdf</file>
  </doc>
  <doc>
    <id>98</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
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    <publisherName/>
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    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1992-12-02</completedDate>
    <publishedDate>1992-12-02</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Packing Steiner Trees: Separation Algorithms.</title>
    <abstract language="eng">In this paper we investigate separation problems for classes of inequalities valid for the polytope associated with the Steiner tree packing problem, a problem that arises, e.~g., in VLSI routing. The separation problem for Steiner partition inequalities is ${\cal N}\hskip-2pt{\cal P}$-hard in general. We show that it can be solved in polynomial time for those instances that come up in switchbox routing. Our algorithm uses dynamic programming techniques. These techniques are also applied to the much more complicated separation problem for alternating cycle inequalities. In this case we can compute in polynomial time, given some point $y$, a lower bound for the gap $\alpha-a^Ty$ over all alternating cycle inequalities $a^Tx\ge\alpha$. This gives rise to a very effective separation heuristic. A by-product of our algorithm is the solution of a combinatorial optimization problem that is interesting in its own right: Find a shortest path in a graph where the ``length'' of a path is its usual length minus the length of its longest edge.</abstract>
    <identifier type="serial">SC-93-02</identifier>
    <identifier type="opus3-id">97</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-984</identifier>
    <identifier type="doi">10.1137/S0895480193258716</identifier>
    <enrichment key="SourceTitle">Appeared in: SIAM J. Discrete Math. 9 (1996) pp. 233-257</enrichment>
    <author>Martin Grötschel</author>
    <author>Alexander Martin</author>
    <author>Robert Weismantel</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-93-02</number>
    </series>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="groetschel">Grötschel, Martin</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/98/SC-93-02.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/98/SC-93-02.pdf</file>
  </doc>
  <doc>
    <id>3105</id>
    <completedYear>1996</completedYear>
    <publishedYear>1996</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>101</pageFirst>
    <pageLast>123</pageLast>
    <pageNumber/>
    <edition/>
    <issue>2</issue>
    <volume>72</volume>
    <type>article</type>
    <publisherName/>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Packing Steiner trees: Polyhedral Investigations</title>
    <parentTitle language="eng">Mathematical Programming, Series A</parentTitle>
    <identifier type="doi">10.1007/BF02592085</identifier>
    <enrichment key="PreprintUrn">urn:nbn:de:0297-zib-786</enrichment>
    <author>Martin Grötschel</author>
    <author>Alexander Martin</author>
    <author>Robert Weismantel</author>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="persons" number="groetschel">Grötschel, Martin</collection>
    <collection role="institutes" number="aopt">Applied Optimization</collection>
  </doc>
  <doc>
    <id>3104</id>
    <completedYear>1996</completedYear>
    <publishedYear>1996</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>23</pageFirst>
    <pageLast>35</pageLast>
    <pageNumber/>
    <edition/>
    <issue>11</issue>
    <volume>31</volume>
    <type>article</type>
    <publisherName>Elsevier</publisherName>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Optimum Path Packing on Wheels</title>
    <parentTitle language="eng">Computers and Mathematics with Applications</parentTitle>
    <enrichment key="PreprintUrn">urn:nbn:de:0297-zib-1976</enrichment>
    <author>Martin Grötschel</author>
    <author>Alexander Martin</author>
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