A Stochastic Probing Problem with Applications

TL;DR

This paper introduces a unified approach to a stochastic probing problem with packing constraints, providing new approximation algorithms and applications to mechanism design.

Contribution

It presents a general stochastic probing framework with matroid and knapsack constraints, unifying and extending prior results, and offers the first polynomial-time approximation for certain mechanism design problems.

Findings

Unified view of stochastic matching and Bayesian mechanism design

First polynomial-time Ω(1/k)-approximate sequential posted price mechanism

Handles complex packing constraints like matroid intersections

Abstract

We study a general stochastic probing problem defined on a universe V, where each element e in V is "active" independently with probability p_e. Elements have weights {w_e} and the goal is to maximize the weight of a chosen subset S of active elements. However, we are given only the p_e values-- to determine whether or not an element e is active, our algorithm must probe e. If element e is probed and happens to be active, then e must irrevocably be added to the chosen set S; if e is not active then it is not included in S. Moreover, the following conditions must hold in every random instantiation: (1) the set Q of probed elements satisfy an "outer" packing constraint, and (2) the set S of chosen elements satisfy an "inner" packing constraint. The kinds of packing constraints we consider are intersections of matroids and knapsacks. Our results provide a simple and unified view of results in stochastic matching and Bayesian mechanism design, and can also handle more general constraints. As an application, we obtain the first polynomial-time $\Omega(1/k)$-approximate "Sequential Posted Price Mechanism" under k-matroid intersection feasibility constraints.