Prophet Inequalities with Linear Correlations and Augmentations

01/28/2020
by   Nicole Immorlica, et al.
0

In a classical online decision problem, a decision-maker who is trying to maximize her value inspects a sequence of arriving items to learn their values (drawn from known distributions), and decides when to stop the process by taking the current item. The goal is to prove a "prophet inequality": that she can do approximately as well as a prophet with foreknowledge of all the values. In this work, we investigate this problem when the values are allowed to be correlated. Since non-trivial guarantees are impossible for arbitrary correlations, we consider a natural "linear" correlation structure introduced by Bateni et al. <cit.> as a generalization of the common-base value model of Chawla et al. <cit.>. A key challenge is that threshold-based algorithms, which are commonly used for prophet inequalities, no longer guarantee good performance for linear correlations. We relate this roadblock to another "augmentations" challenge that might be of independent interest: many existing prophet inequality algorithms are not robust to slight increase in the values of the arriving items. We leverage this intuition to prove bounds (matching up to constant factors) that decay gracefully with the amount of correlation of the arriving items. We extend these results to the case of selecting multiple items by designing a new (1+o(1)) approximation ratio algorithm that is robust to augmentations.

READ FULL TEXT

page 1

page 2

page 3

page 4

11/29/2018

Smoothed Analysis of Multi-Item Auctions with Correlated Values

Consider a seller with m heterogeneous items for sale to a single additi...
08/10/2022

Knapsack Secretary Through Boosting

We revisit the knapsack-secretary problem (Babaioff et al.; APPROX 2007)...
04/04/2022

Order Selection Prophet Inequality: From Threshold Optimization to Arrival Time Design

In the classical prophet inequality, a gambler faces a sequence of items...
06/21/2021

On Simple Mechanisms for Dependent Items

We study the problem of selling n heterogeneous items to a single buyer,...
10/30/2017

Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the fie...
11/18/2020

Non-Adaptive Matroid Prophet Inequalities

We investigate non-adaptive algorithms for matroid prophet inequalities....
10/19/2020

Constrained-Order Prophet Inequalities

Free order prophet inequalities bound the ratio between the expected val...