Thrifty Algorithms for Multistage Robust Optimization

摘要

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(log m + log n)-approximation algorithm, nearly matching the Ω (log n + log m/log log m) 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, log n, log λ_(max)}), where λ_(max) is the maximum inflation over all the stages. We conjecture that these problems also admit O(1)-approximate thrifty algorithms.