Solving k-cluster problems to optimality with semidefinite programming

Jérôme Malick (1) , Frédéric Roupin (2)

1 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
2 LIPN - Laboratoire d'Informatique de Paris-Nord

This paper deals with the computation of exact solutions of a classical NP-hard problem in combinatorial optimization, the k-cluster problem. This problem consists in finding a heaviest subgraph with k nodes in an edge weighted graph. We present a branch-and-bound algorithm that applies a novel bounding procedure, based on recent semidefinite programming techniques. We use new semidefinite bounds that are less tight than the standard semidefinite bounds, but cheaper to get. The experiments show that this approach is competitive with the best existing ones.

Mots clés: combinatorial optimization; k-cluster; exact resolution; semidefinite programming; Lagrangian relaxation; branch-and-bound

Domaines: Optimisation et contrôle [math.OC]; Recherche opérationnelle [math.OC]; Mathématique discrète [cs.DM]

Jérôme Malick, Frédéric Roupin. Solving k-cluster problems to optimality with semidefinite programming. Mathematical Programming, 2012, 136 (2), pp.279-300. ⟨10.1007/s10107-012-0604-1⟩. ⟨hal-00609744⟩