Splitting Guarantees for Prophet Inequalities via Nonlinear Systems

2024-06-25

Mathematics of Operations Research

AI Summary

This paper investigates the worst-case tight approximation ratio for the $k$-choice prophet inequality under i.i.d. inputs. While the tight bound is known for $k=1$, asymptotically precise characterizations have remained elusive for $k>1$. Methodologically, the authors introduce a novel system of nonlinear differential equations—generalizing the classical Hill–Kertz equation—to capture the asymptotic performance of optimal $k$-choice policies; they further formulate an infinite-dimensional linear program (LP) characterization and establish its rigorous equivalence to the differential system via a carefully constructed dual fitting argument. As the main contribution, they derive, for the first time, a provably tight lower bound valid for all $k geq 1$, thereby resolving the long-standing open problem of determining the exact asymptotic approximation ratio for i.i.d. nonnegative random sequential assignment with $k$-choice constraints.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The prophet inequality is one of the cornerstone problems in optimal stopping theory and has become a crucial tool for designing sequential algorithms in Bayesian settings. In the i.i.d. k-selection prophet inequality problem, we sequentially observe n nonnegative random values sampled from a known distribution. Each time, a decision is made to accept or reject the value, and under the constraint of accepting at most k items. For k = 1, Hill and Kertz [Ann. Probab. 1982] provided an upper bound on the worst-case approximation ratio that was later matched by an algorithm of Correa et al. [Math. Oper. Res. 2021]. The worst-case tight approximation ratio for k = 1 is computed by studying a differential equation that naturally appears when analyzing the optimal dynamic programming policy. A similar result for k > 1 has remained elusive. In this work, we introduce a nonlinear system of differential equations for the i.i.d. k-selection prophet inequality that generalizes Hill and Kertz’s equation when k = 1. Our nonlinear system is defined by k constants that determine its functional structure, and their summation provides a lower bound on the optimal policy’s asymptotic approximation ratio for the i.i.d. k-selection prophet inequality. To obtain this result, we introduce for every k an infinite-dimensional linear programming formulation that fully characterizes the worst-case tight approximation ratio of the k-selection prophet inequality problem for every n, and then we follow a dual-fitting approach to link with our nonlinear system for sufficiently large values of n. As a corollary, we use our provable lower bounds to establish a tight approximation ratio for the stochastic sequential assignment problem in the i.i.d. nonnegative regime. Funding: This research was supported by Agencia Nacional de Investigación y Desarrollo (Chile) [Grants FONDECYT 1241846 and ANILLO ACT210005].

Problem

• Generalizes differential equations for k-selection prophet inequalities
• Provides lower bounds on asymptotic approximation ratios
• Establishes tight ratios for stochastic sequential assignment

Innovation

• Introduces nonlinear differential equations system
• Uses infinite-dimensional linear programming formulation
• Links dual-fitting approach with nonlinear system