Conceitos Básicos
This work introduces sharding and Poissonization as a unified framework for analyzing and improving upon prophet inequalities. The framework simplifies proofs and refines the competitive ratio analysis of several well-studied prophet inequalities.
Resumo
The key ideas in this work are:
Poissonization:
Modeling the random variables using Poisson distributions to simplify probability calculations.
Showing that as the number of "shards" (subdivisions) of each random variable goes to infinity, the distribution of the shards can be approximated by a Poisson distribution.
Sharding:
Splitting each random variable into multiple independent "shards" that collectively mimic the original variable's behavior.
Using the Poisson approximation of the shards to bound the competitive ratio of various prophet inequality algorithms.
The framework is applied to improve the analysis of several prophet inequality problems:
Top-1-of-k prophet inequality:
Significantly improves the lower and upper bounds for small values of k (e.g., k=2,3,4).
Provides a new asymptotic bound for general k that is tighter than previous results.
Prophet secretary problem:
Raises the lower bound on the competitive ratio from 0.669 to 0.6724.
IID Semi-Online prophet inequality:
Improves the lower bound on the competitive ratio from 0.869 to 0.89.
Semi-Online-Load-Minimization (SOLM) problem:
Achieves a 1-o(1) competitive ratio with O(log*n) load, improving the previous O(log n) bound.
The framework also provides simpler proofs for several known results in the literature, unifying the analysis.