Approximating the shortest path problem with scenarios

This paper discusses the shortest path problem in a general directed graph with n nodes and K cost scenarios (objectives). In order to choose a solution, the min-max criterion is applied. The min-max version of the problem is hard to approximate within $\Omega ({\log }^{1-ϵ}K)$ for any $ϵ>0$ unless NP $\subseteq \text{DTIME}\left(n^{\text{polylog}n}\right)$ even for arc series-parallel graphs and within $\Omega (\log n/\log \log n)$ unless NP $\subseteq \text{ZPTIME}\left(n^{\log \log n}\right)$ for acyclic graphs. The best approximation algorithm for the min-max shortest path problem in general graphs, known to date, has an approximation ratio of K. In this paper, an $\tilde{O}\left(\sqrt{n}\right)$ flow LP-based approximation algorithm for min-max shortest path in general graphs is constructed. It is also shown that the approximation ratio obtained is close to an integrality gap of the corresponding flow LP relaxation.