Pivot Rules for Circuit-Augmentation Algorithms in Linear Optimization

TL;DR

This paper investigates circuit-augmentation algorithms in linear optimization, revealing complexity results, polynomial bounds, and novel insights into the Simplex method, especially for 0/1-LPs and polyhedra.

Contribution

It establishes NP-hardness of certain pivot rules, provides polynomial bounds for circuit-diameter, and links circuit-augmentation to classical Simplex path problems.

Findings

Greatest-improvement and Dantzig rules are NP-hard for 0/1-LPs.

Steepest-descent rule is polynomial-time computable and strongly-polynomial for 0/1-LPs.

Circuit-diameter is polynomially bounded by input size for any polyhedron.

Abstract

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