Exponential Lower Bounds for History-Based Simplex Pivot Rules on Abstract Cubes

The behavior of the simplex algorithm is a widely studied subject. Specifically, the question of the existence of a polynomial pivot rule for the simplex algorithm is of major importance. Here, we give exponential lower bounds for three history-based pivot rules. Those rules decide their next step based on memory of the past steps. In particular, we study Zadeh’s least entered rule, Johnson’s least-recently basic rule and Cunningham’s least-recently considered (or round-robin) rule. We give exponential lower bounds on Acyclic Unique Sink Orientations (AUSO) of the abstract cube, for all of these pivot rules. For Johnson’s rule our bound is the first superpolynomial one in any context; for Zadeh’s it is the first one for AUSO. Those two are our main results.