Pivot Rules for Circuit-Augmentation Algorithms in Linear Optimization

Circuit-augmentation algorithms are generalizations of the simplex method, where in each step one is allowed to move along a fixed set directions, called circuits, that superset edges polytope. We show circuit-augmentation framework greatest-improvement and Dantzig pivot rules NP-hard, already for 0/1-LPs. Differently, steepest-descent rule can be carried out polynomial time 0/1 setting, number circuit augmentations required reach an optimal solution according this strongly The has been interest as proxy steps circuit-diameter polyhedra studied lower bound combinatorial diameter polyhedra. Extending prior results, we any polyhedron $P$ bounded by input bit-size $P$. This contrast with best bounds Interestingly, exploited make novel conclusions about classical method itself: In particular, byproduct our prove (i) computing shortest (monotone) path on 1-skeleton polytope hard approximate within factor better than 2, (ii) $0/1$ polytopes, monotone length constructed using steepest improving edges.