🤖 AI Summary
This work addresses the long-standing bottleneck in prophet inequality research—manual construction of worst-case instances for tight competitive ratio proofs—by proposing the first unified framework that directly models tightness analysis as a computationally tractable optimization problem. Methodologically, it formalizes worst-case instance search as a convex optimization and linear programming problem subject to extremal probability distribution constraints, integrating tools from random-order theory and extremal probability analysis, thereby replacing the traditional decoupled paradigm of “algorithm analysis + counterexample construction.” Contributions include: (i) the first automated computation of tight competitive ratios; (ii) a unified derivation of tight bounds for multiple prophet inequality variants, yielding several new results; and (iii) rigorous verification of the optimality of several classical bounds. The framework significantly enhances the systematicity, scalability, and reliability of tightness proofs in online stochastic optimization.
📝 Abstract
Prophet inequalities consist of many beautiful statements that establish tight performance ratios between online and offline allocation algorithms. Typically, tightness is established by constructing an algorithmic guarantee and a worst-case instance separately, whose bounds match as a result of some "ingenuity". In this paper, we instead formulate the construction of the worst-case instance as an optimization problem, which directly finds the tight ratio without needing to construct two bounds separately.
