The Strongest Facets of the Acyclic Subgraph Polytope Are Unknown

Abstrac t . We consider the acyclic subgraph polytope and define the notion of strength of a relaxation as the maximum improvement obtained by using this relaxation instead of the most trivial relaxation of the problem. We show that the strength of a relaxation is the maximum of the strengths of the relaxations obtained by simply adding to the trivial relaxation each valid inequality separately. We also derive from the probabilistic method that the maximum strength of any inequality is 2. We then consider all (or almost all) the known valid inequalities for the polytope and compute their strength. The surprising observation is that their strength is at most slightly more than 3/2, implying that the strongest inequalities are yet unknown. We then consider a pseudo-random construction due to Alon and Spencer based on quadratic residues to obtain new facet-defining inequalities for the polytope. These are also facetdefining for the linear ordering polytope.