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Dantzig-Wolfe Reformulations for the Stable Set Problem

Dantzig-Wolfe reformulation of an integer program convexifies a subset of the constraints, which yields an extended formulation with a potentially stronger linear programming (LP) relaxation than the original formulation. This paper is part of an endeavor to understand the strength of such a reformulation in general. We investigate the strength of Dantzig-Wolfe reformulations of the classical edge formulation for the maximum weighted stable set problem. Since every constraint in this model corresponds to an edge of the underlying graph, a Dantzig-Wolfe reformulation consists of choosing a subgraph and convexifying all constraints corresponding to edges of this subgraph. We characterize Dantzig-Wolfe reformulations not yielding a stronger LP relaxation (than the edge formulation) as reformulations where this subgraph is bipartite. Furthermore, we analyze the structure of facets of the stable set polytope and present a characterization of Dantzig-Wolfe reformulations with the strongest possible LP relaxation as reformulations where the chosen subgraph contains all odd holes (and 3-cliques). To the best of our knowledge, these are the first non-trivial general results about the strength of relaxations obtained from decomposition methods, after Geoffrion’s seminal 1974 paper about Lagrangian relaxation.