On the circuit diameter conjecture for counterexamples to the Hirsch conjecture

Circuit diameters of polyhedra are a fundamental tool for studying the complexity of circuit augmentation schemes for linear programming and for finding lower bounds on combinatorial diameters. The main open problem in this area is the circuit diameter conjecture, the analogue of the Hirsch conjecture in the circuit setting. A natural question is whether the well-known counterexamples to the Hirsch conjecture carry over. Previously, Stephen and Yusun showed that the Klee-Walkup counterexample to the unbounded Hirsch conjecture does not transfer to the circuit setting. Our main contribution is to show that the original counterexamples for other variants, using monotone walks or for bounded polytopes, also do not transfer. A challenge lies in the dependence of circuit diameters on the specific realization of a polyhedron. We discuss for which realizations, in addition to the original ones from the literature, our tools resolve this question.

Our results rely on new observations on structural properties of these counterexamples. To analyze the bounded case, we exploit the geometry of certain 2-faces of the polytopes underlying all known bounded Hirsch counterexamples in Santos’ work. For Todd’s monotone Hirsch counterexample, we study linear programs on spindles and prove sufficient conditions for short monotone circuit walks to exist. We then enumerate all linear programs over Todd’s polytope and find four new orientations that contradict the monotone Hirsch conjecture, while the remaining 7107 satisfy the bound. The conclusion then follows by applying these sufficient conditions to Todd’s counterexample.