Computational complexity of the recoverable robust shortest path problem in acyclic digraphs under interval budgeted uncertainty

Abstract

In this paper, we consider the recoverable robust shortest path problem in acyclic digraphs, employing interval budgeted uncertainty to model uncertain second-stage costs. For the continuous budgeted uncertainty model, we prove that the problem is strongly NP-hard even in layered digraphs. Furthermore, we show that in general acyclic digraphs, the problem cannot be approximated within any constant factor unless $\mathrm{P}=\mathrm{NP}$, nor can it be approximated within a factor of $2^{{\log }^{1-\epsilon }n}$ for any $\epsilon >0$ unless $\mathrm{NP}\subseteq \mathrm{DTIME}(n^{\mathrm{poly}\log n})$. For the discrete budgeted uncertainty model, we show that the problem is not approximable unless $\mathrm{P}=\mathrm{NP}$, even in layered digraphs. Finally, we establish that under continuous budgeted uncertainty, the integrality gap of a relaxation allowing a fractional first-stage solution is at least $\Omega (\sqrt{n})$.