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

TL;DR

This work tackles Matroid Basis Testing (MBT) under partition matroids within the SBFE framework, aiming to minimize adaptive query costs while certifying the existence of a basis of active elements. The authors introduce a novel budget-in-expectation problem MP0 and design an O(1)-approximation algorithm (ALG) for MBT on partition matroids by coupling a 1-certificate strategy (OPT${}_1$) with a 0-certificate strategy (SteepestAscent) and by solving MP0-typed subproblems using an InsideOut (IO) approach. The key technical contributions include a phase-based algorithm that allocates budgets across partition classes (via a continuous knapsack argument), a polynomial-time method to approximate MP0 on uniform/partition matroids through randomized pruning of IO, and a suite of lemmas (e.g., TableKeyLemma, IOapprox, IOtreesurgery) that establish approximation guarantees and polynomial-time computability. The results offer a robust framework for MBT on partition matroids and open avenues for applying budget-in-expectation concepts to broader SBFE problems and stochastic combinatorial optimization. The findings advance both theory and potential applications in adaptive querying under stochastic constraints, with implications for efficient testing in complex combinatorial structures.

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.

Figures (13)

• Figure 1: Overview of the results and how they interact with each other.
• Figure 2: Left: the instance on which depth-first strategies and non-adaptive strategies perform poorly (notation: "variable name: probability"). Right: the optimal strategy for that instance.
• Figure 3: Left: A uniform matroid (a partition class from the counterexample for depth-first and non-adaptive strategies). Right: A (not-to-scale) graph of the function relating budget and attainable probability of finding a $0$-certificate for that uniform matroid, along with the corresponding strategies. The red octagon corresponds to stopping; when a "?" is depicted, it only corresponds to stopping with a certain probability. The (randomized) strategy on the left is for budgets between 0 and 2, and the strategy on the right is for budgets between 2 and approximately $\frac{1}{\varepsilon^2}$.
• Figure 4: Case 1.1 (Continue-Continue). $S_j$ on the left, $S'_j$ on the right (note swapped roles of $T^{01}$ and $T^{10}$ so that these subtrees are reached with the same probability as before).
• Figure 5: Case 1.2 (Continue-Abort). $S_j$ on the left; $S'_j$ on the right, defined in the text as random selection over two deterministic strategies ($S"_j$ and $\tilde{S}"_j$), and can be depicted in this way. The octagonal nodes correspond to stopping without a 0-certificate (with certainty or with probability $1-\alpha$ and continuing otherwise). In this case, $\alpha := \frac{\bar{p}_{i} - \bar{p}_{a_1}}{\bar{p}_{i} \cdot \mathopen{}\mathclose{\left( {1-\bar{p}_{a_1}} \right)}}$.

Theorems & Definitions (21)

• Lemma 1
• Lemma 2
• Theorem 1
• Lemma 3
• Lemma 4
• Lemma 5
• proof
• proof : Proof of \ref{['thm:maintheorem']}
• Lemma 5
• proof : Proof of \ref{['OPT1BOptimal']}