A Lagrangian relaxation approach to the edge-weighted clique problem

The b-clique polytope CP 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 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 . The problem of optimizing linear functions over CP b has so far been approached via heuristic combinatorial algorithms and cutting-plane methods. We study the structure of CP 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.