A Lagrangian relaxation approach to the edge-weighted clique problem

M.M.G. Hunting, U. Faigle, Walter Kern

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The $b$-clique polytope $CP^n_b$ is the convex hull of the node and edge incidence vectors of all subcliques of size at most $b$ of a complete graph on $n$ nodes. Including the Boolean quadric polytope $QP^n$ as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over $QP^n_n$. The problem of optimizing linear functions over $CP^n_b$ has so far been approached via heuristic combinatorial algorithms and cutting-plane methods. We study the structure of $CP^n_b$ in further detail and present a new computational approach to the linear optimization problem based on Lucena's suggestion of integrating cutting planes into a Lagrangian relaxation of an integer programming problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented.
Original languageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente
Number of pages21
ISBN (Print)0169-2690
Publication statusPublished - 1998

Publication series

NameMemorandum / Faculty of Mathematical Sciences
PublisherDepartment of Applied Mathematics, University of Twente
ISSN (Print)0169-2690


  • EWI-3296
  • MSC-90C27
  • METIS-141301
  • IR-65665
  • MSC-90D12

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