Pivot Rules for Circuit-Augmentation Algorithms in Linear Optimization
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Circuit-augmentation algorithms are a generalization of the Simplex method, where in each step one is allowed to move along a set of directions, called circuits, that is a superset of the edges of a polytope. We show that in this general context the greatest-improvement and Dantzig pivot rules are NP-hard. Differently, the steepest-descent pivot rule can be computed in polynomial time, and the number of augmentations required to reach an optimal solution according to this rule is strongly-polynomial for 0/1 LPs. Interestingly, we show that this more general framework can be exploited also to make conclusions about the Simplex method itself. In particular, as a byproduct of our results, we prove that (i) computing the shortest monotone path to an optimal solution on the 1-skeleton of a polytope is NP-hard, and hard to approximate within a factor better than 2, and (ii) for 0/1 polytopes, a monotone path of polynomial length can be constructed using steepest improving edges.
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