Thrifty Algorithms for Multistage Robust Optimization

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We consider a class of multi-stage robust covering problems, where additional information is revealed about the problem instance in each stage, but the cost of taking actions increases. The dilemma for the decision-maker is whether to wait for additional information and risk the inflation, or to take early actions to hedge against rising costs. We study the “k-robust” uncertainty model: in each stage i = 0, 1, …, T, the algorithm is shown some subset of size k i that completely contains the eventual demands to be covered; here k 1 > k 2 > ⋯ > k T which ensures increasing information over time. The goal is to minimize the cost incurred in the worst-case possible sequence of revelations.

For the multistage k-robust set cover problem, we give an O(logm + logn)-approximation algorithm, nearly matching the \(\Omega\left(\log n+\frac{\log m}{\log\log m}\right)\) hardness of approximation [4] even for T = 2 stages. Moreover, our algorithm has a useful “thrifty” property: it takes actions on just two stages. We show similar thrifty algorithms for multi-stage k-robust Steiner tree, Steiner forest, and minimum-cut. For these problems our approximation guarantees are O( min { T, logn, logλ max }), where λ max is the maximum inflation over all the stages. We conjecture that these problems also admit O(1)-approximate thrifty algorithms.