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

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.