A Note on the Approximability of Deepest-Descent Circuit Steps
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Linear programs (LPs) can be solved through a polynomial number of so-called deepest-descent circuit steps, each step following a circuit direction to a new feasible solution of most-improved objective function value. A computation of deepest-descent steps has recently been shown to be NP-hard [De Loera et al., arXiv, 2019]. This is a consequence of the hardness of what we call the optimal circuit-neighbor problem (OCNP) for LPs with non-unique optima. However, the non-uniqueness assumption is crucial to the hardness of OCNP, because we show that OCNP for LPs with a unique optimum is solvable in polynomial time. Moreover, in practical applications one is usually only interested in finding some optimum of an LP, in which case a simple perturbation of the objective yields an instance with a unique optimum. It is thus natural to ask whether deepest-descent steps are also easy to compute for LPs with unique optima, or whether this problem is hard despite OCNP being easy. We show that deepest-descent steps can be efficiently approximated within a factor of n, where n is the dimension of the polyhedron at hand, but not within a factor of O(n^1-ϵ) for any ϵ > 0. While we prove that OCNP can be solved efficiently for LPs with a unique optimum, our different hardness approach allows us to show strong negative results: computing deepest-descent steps is NP-hard and inapproximable even for 0/1 linear programs with a unique optimum which are defined by a totally unimodular constraint matrix.
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