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.
翻译:线性程序(LPs)可以通过多步的多步移动在改善电路方向上改善目标的路径,即所谓的最深白步骤(d-d-steps)来解决。计算这些步骤是NP-hard(De Loera et al., arXiv, 2019),其原因是在非单一的opima 的LPs上决定最佳路路段邻居(OCNP)的难度很大。我们证明OCNP在独特的opima 的许诺下很容易,但已经是 $(n ⁇ 1-\ varepsilan} $(n ⁇ 1-\ varepsilan} $-coprocipl-d-steps d-steps 即使在完全不单单面的 $n$- expion 0/1-LPs 和独特的最佳情况下,我们提供匹配的 $n- approximation。