A Lagrangian relaxation approach to the edge-weighted clique problem

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.