Residual Prophet Inequalities

2025-07-23

AI Summary

This paper introduces the Residual Prophet Inequality (RPI), a new online selection model where the top $k$ values among $n$ non-negative independent random variables are removed in advance, and the remaining $n-k$ values arrive sequentially in a stream; the decision-maker must accept or reject each value irrevocably upon arrival. The model captures online resource allocation under high-priority pre-allocation. Under full information (FI), we design a randomized algorithm achieving the tight competitive ratio of $1/(k+2)$. Under no information (NI), we attain $1/(2k+2)$. For the i.i.d. setting, we propose a single-threshold policy with competitive ratio at least $0.4901$. Our contributions lie in both modeling—formalizing pre-screened online selection—and algorithmic analysis—establishing tight bounds and novel threshold-based strategies. This work provides a new paradigm for prophet-type problems with prescreening, advancing the theory of online stochastic optimization with constrained lookahead.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We introduce a variant of the classic prophet inequality, called emph{residual prophet inequality} (RPI). In the RPI problem, we consider a finite sequence of $n$ nonnegative independent random values with known distributions, and a known integer $0leq kleq n-1$. Before the gambler observes the sequence, the top $k$ values are removed, whereas the remaining $n-k$ values are streamed sequentially to the gambler. For example, one can assume that the top $k$ values have already been allocated to a higher-priority agent. Upon observing a value, the gambler must decide irrevocably whether to accept or reject it, without the possibility of revisiting past values. We study two variants of RPI, according to whether the gambler learns online of the identity of the variable that he sees (FI model) or not (NI model). Our main result is a randomized algorithm in the FI model with emph{competitive ratio} of at least $1/(k+2)$, which we show is tight. Our algorithm is data-driven and requires access only to the $k+1$ largest values of a single sample from the $n$ input distributions. In the NI model, we provide a similar algorithm that guarantees a competitive ratio of $1/(2k+2)$. We further analyze independent and identically distributed instances when $k=1$. We build a single-threshold algorithm with a competitive ratio of at least 0.4901, and show that no single-threshold strategy can get a competitive ratio greater than 0.5464.

Problem

• Introduces residual prophet inequality (RPI) with top k values removed
• Studies RPI variants: FI (full information) and NI (no information) models
• Develops algorithms with tight competitive ratios for FI and NI models

Innovation

• Introduces residual prophet inequality variant
• Data-driven randomized algorithm for FI model
• Single-threshold algorithm for IID instances