On the approximability of minmax (regret) network optimization problems

TL;DR

This paper investigates the difficulty of approximating minmax and regret versions of basic network problems, showing they are hard to approximate when the number of scenarios is unbounded, unless certain complexity class inclusions hold.

Contribution

It demonstrates the inapproximability bounds for minmax and regret network problems with unbounded scenarios, extending understanding of their computational complexity.

Findings

Problems are not approximable within log^{1-\u03b5} K for any psilon>0

Inapproximability holds unless NP TIME(n^{poly log n})

Results apply to basic polynomially solvable network problems

Abstract

In this paper the minmax (regret) versions of some basic polynomially solvable deterministic network problems are discussed. It is shown that if the number of scenarios is unbounded, then the problems under consideration are not approximable within $\log^{1-\epsilon} K$ for any $\epsilon>0$ unless NP $\subseteq$ DTIME$(n^{\mathrm{poly} \log n})$, where $K$ is the number of scenarios.