A Lagrangian relaxation approach to the edge-weighted clique problem

Hunting, Marcel and Faigle, Ulrich and Kern, Walter (2001) A Lagrangian relaxation approach to the edge-weighted clique problem. European Journal of Operational Research, 131 (1). pp. 119-131. ISSN 0377-2217

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Abstract:	The b-clique polytope CPnb 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 QPn=CPnn 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 CPnn. The problem of optimizing linear functions over CPnb has so far been approached via heuristic combinatorial algorithms and cutting-plane methods. We study the structure of CPnb in further detail and present a new computational approach to the linear optimization problem based on the idea of integrating cutting planes into a Lagrangian relaxation of an integer programming problem that Balas and Christofides had suggested for the traveling salesman 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:	Article
Copyright:	© 2001 Elsevier
Faculty:	Electrical Engineering, Mathematics and Computer Science (EEMCS)
Research Group:	Discrete Mathematics and Mathematical Programming (DMMP)
Link to this item:	http://purl.utwente.nl/publications/74474
Official URL:	http://dx.doi.org/10.1016/S0377-2217(99)00449-X
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