Approximating some network problems with scenarios

In this paper the shortest path and the minimum spanning tree problems in a graph with $n$ nodes and $K$ cost scenarios (objectives) are discussed. In order to choose a solution the min-max criterion is applied. The minmax versions of both problems are hard to approximate within $O(\log^{1-\epsilon} K)$ for any $\epsilon>0$. The best approximation algorithm for the min-max shortest path problem, known to date, has approximation ratio of $K$. On the other hand, for the min-max spanning tree, there is a randomized algorithm with approximation ratio of $O(\log² n)$. In this paper a deterministic $O(\sqrt{n\log K/\log\log K})$-approximation algorithm for min-max shortest path is constructed. For min-max spanning tree a deterministic $O(\log n \log K/\log\log K)$-approximation algorithm is proposed, which works for a large class of graphs and a randomized $O(\log n)$-approximation algorithm, which can be applied to all graphs, is constructed. It is also shown that the approximation ratios obtained are close to the integrality gaps of the corresponding LP relaxations.