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A Lagrangian relaxation approach to the edge-weighted clique problem

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Hunting, Marcel and Faigle, Ulrich and Kern, Walter (1998) A Lagrangian relaxation approach to the edge-weighted clique problem. [Report]

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Abstract: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.
Item Type:Report
Faculty:
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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Link to this item:http://purl.utwente.nl/publications/65665
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