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This paper discusses the shortest path problem in a general directed graphwith $n$ nodes and $K$ cost scenarios (objectives). In order to choose asolution, the min-max criterion is applied. The min-max version of the problemis hard to approximate within $\Omega(\log^{1-\epsilon} K)$ for any$\epsilon>0$ unless NP$\subseteq \text{DTIME}(n^{\text{polylog} \,n})$ even forarc series-parallel graphs and within $\Omega(\log n/\log\log n)$ unlessNP$\subseteq \text{ZPTIME}(n^{\log\log n})$ for acyclic graphs. The bestapproximation algorithm for the min-max shortest path problem in generalgraphs, known to date, has an approximation ratio of~$K$. In this paper, an$\widetilde{O}(\sqrt{n})$ flow LP-based approximation algorithm for min-maxshortest path in general graphs is constructed. It is also shown that theapproximation ratio obtained is close to an integrality gap of thecorresponding flow LP relaxation.

Approximating the shortest path problem with scenarios