Approximating Matroid Basis Testing for Partition Matroids using Budget-In-Expectation

2026-01-10

arXiv.org

AI Summary

This study addresses the adaptive evaluation of random Boolean functions over partition matroids: given a ground set whose elements have unknown active states, the goal is to determine whether there exists a basis consisting entirely of active elements, using the minimum expected number of queries. To this end, the work proposes a novel approach that integrates adaptive randomized strategies, optimization under expected budget constraints, and interleaved solving across multiple instances, yielding the first polynomial-time constant-factor approximation algorithm for this problem. This result overcomes the limitation of prior methods, which lacked provable approximation guarantees, and establishes an effective algorithmic framework for stochastic query problems with expected-cost constraints.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We consider the following Stochastic Boolean Function Evaluation problem, which is closely related to several problems from the literature. A matroid $\mathcal{M}$ (in compact representation) on ground set $E$ is given, and each element $i\in E$ is active independently with known probability $p_i\in(0,1)$. The elements can be queried, upon which it is revealed whether the respective element is active or not. The goal is to find an adaptive querying strategy for determining whether there is a basis of $\mathcal{M}$ in which all elements are active, with the objective of minimizing the expected number of queries. When $\mathcal{M}$ is a uniform matroid, this is the problem of evaluating a $k$-of-$n$ function, first studied in the 1970s. This problem is well-understood, and has an optimal adaptive strategy that can be computed in polynomial time. Taking $\mathcal{M}$ to instead be a partition matroid, we show that previous approaches fail to give a constant-factor approximation. Our main result is a polynomial-time constant-factor approximation algorithm producing a randomized strategy for this partition matroid problem. We obtain this result by combining a new technique with several well-established techniques. Our algorithm adaptively interleaves solutions to several instances of a novel type of stochastic querying problem, with a constraint on the $\textit{expected}$ cost. We believe that this type of problem is of independent interest, will spark follow-up work, and has the potential for additional applications.

Problem

• matroid basis testing
• partition matroid
• stochastic Boolean function evaluation
• adaptive querying
• expected query cost

Innovation

• partition matroid
• stochastic Boolean function evaluation
• constant-factor approximation
• budget-in-expectation
• adaptive querying strategy