  Linear programs (LPs) can be solved by polynomially many moves along the
circuit direction improving the objective the most, so-called deepest-descent
steps (dd-steps). Computing these steps is NP-hard (De Loera et al., arXiv,
2019), a consequence of the hardness of deciding the existence of an optimal
circuit-neighbor (OCNP) on LPs with non-unique optima.
  We prove OCNP is easy under the promise of unique optima, but already
$O(n^{1-\varepsilon})$-approximating dd-steps remains hard even for totally
unimodular $n$-dimensional 0/1-LPs with a unique optimum. We provide a matching
$n$-approximation.
