# (Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing

## TL;DR

This paper studies the adaptivity gap in stochastic multi-value probing problems, providing near-optimal bounds for various functions and constraints, thereby advancing understanding of non-adaptive strategies in complex probabilistic settings.

## Contribution

It introduces a multi-value stochastic probing framework and establishes tight bounds on the adaptivity gap for key classes of functions and constraints, resolving open questions.

## Key findings

- Adaptivity gap at most 2 for monotone submodular functions.
- Adaptivity gap between O(k log k) and k for weighted rank functions of k-extendible systems.
- Results extend previous Bernoulli case bounds to multi-value distributions.

## Abstract

Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge $e$ exists, we need to perform an expensive test, in which case the edge $e$ appears independently with a \emph{known} probability $p_e$. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size. The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is \emph{adaptive}---decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the \emph{adaptivity gap} is small, i.e., the best \emph{non-adaptive} strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values. Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most $2$ and provide a matching lower bound. For a weighted rank function of a $k$-extendible system (a generalization of intersection of $k$ matroids), we show the adaptivity gap is between $O(k\log k)$ and $k$. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al.