A Note on the Approximability of Deepest-Descent Circuit Steps

TL;DR

This paper investigates the computational complexity of approximating deepest-descent steps in linear programming, showing that even under certain conditions, approximating these steps remains computationally hard, with some cases admitting a tight approximation.

Contribution

It proves that approximating deepest-descent steps is hard even for specific LPs, and provides a matching n-approximation algorithm.

Findings

Approximating dd-steps is NP-hard even for totally unimodular 0/1-LPs with a unique optimum.

OCNP is easier under the promise of unique optima.

A matching n-approximation algorithm for dd-steps is presented.

Abstract

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.