Submodular Stochastic Probing with Prices

TL;DR

This paper introduces Stochastic Probing with Prices (SPP), a new model where probing elements incurs costs, and provides approximation algorithms for online scenarios, advancing the understanding of stochastic probing problems.

Contribution

The paper formulates SPP, a novel variant of stochastic probing with costs, and develops bi-criteria approximation algorithms for the online version.

Findings

Provides a bi-criteria approximation algorithm for online SPP.

Achieves state-of-the-art approximations for traditional stochastic probing.

Extends stochastic probing models to include pricing and cost considerations.

Abstract

We introduce Stochastic Probing with Prices (SPP), a variant of the Stochastic Probing (SP) model in which we must pay a price to probe an element. A SPP problem involves two set systems $(N,\mathcal{I}_{in})$ and $(N,\mathcal{I}_{out})$ where each $e\in N$ is active with probability $p_e$. To discover whether $e$ is active, it must be probed by paying the price $\Delta_e$. If it is probed and active, then it is irrevocably added to the solution. Moreover, at all times, the set of probed elements must lie in $\mathcal{I}_{out}$, and the solution (the set of probed and active elements) must lie in $\mathcal{I}_{in}$. The goal is to maximize a set function $f$ minus the cost of the probes. We give a bi-criteria approximation algorithm to the online version of this problem, in which the elements are shown to the algorithm in a possibly adversarial order. Our results translate to state-of-the-art approximations for the traditional (online) stochastic probing problem.