Stochastic Function Certification with Correlations

Abstract: We study the Stochastic Boolean Function Certification (SBFC) problem, where we are given $n$ Bernoulli random variables ${X_e: e \in U}$ on a ground set $U$ of $n$ elements with joint distribution $p$, a Boolean function $f: 2^U \to {0, 1}$, and an (unknown) scenario $S = {e \in U: X_e = 1}$ of active elements sampled from $p$. We seek to probe the elements one-at-a-time to reveal if they are active until we can certify $f(S) = 1$, while minimizing the expected number of probes. Unlike most previous results that assume independence, we study correlated distributions $p$ and give approximation algorithms for several classes of functions $f$. When $f(S)$ is the indicator function for whether $S$ is the spanning set of a given matroid, our problem reduces to finding a basis of active elements of a matroid by probing elements. We give a non-adaptive $O(\log n)$-approximation algorithm for arbitrary distributions $p$, and show that this is tight up to constants unless P $=$ NP, even for partition matroids. For uniform matroids, we give constant factor $4.642$-approximation ([BBFT20]) that can be further improved to a $2$-approximation if additionally the random variables are negatively correlated for the case of $1$-uniform matroid. We also give an adaptive $O(\log k)$-approximation algorithm for SBFC for $k$-uniform matroids for the Graph Probing problem, where we seek to probe the edges of a graph one-at-a-time until we find $k$ active edges. The underlying distribution on edges arises from (hidden) independent vertex random variables, with an edge being active if at least one of its endpoints is active. This significantly improves over the information-theoretic lower bound on $Ω(\mathrm{poly}(n))$ ([JGM19]) for adaptive algorithms for $k$-uniform matroids with arbitrary distributions.

• The paper’s core contribution is developing tight O(log n) and constant-factor approximation algorithms for matroid and k-uniform certifications under correlated uncertainty.
• It employs advanced LP relaxations with knapsack cover inequalities and randomized matroid-based rounding to efficiently manage exponentially many constraints.
• Adaptive probing strategies, including graph and hypergraph formulations, significantly reduce adaptivity gaps and enhance certification performance.

Summary of "Stochastic Function Certification with Correlations" (2604.02611)

Problem Formulation and Motivation

The paper addresses the Stochastic Boolean Function Certification (SBFC) problem, which generalizes stochastic combinatorial evaluation tasks under correlated uncertainty. Given a collection of $n$ Bernoulli random variables $\{X_e\}$ over a universe $U$ with possibly correlated joint distribution $p$, and a monotone Boolean function $f: 2^U \to \{0, 1\}$, the goal is to probe elements sequentially until one can certify $f(S) = 1$ for the realized active subset $S$, minimizing expected probe count. Unlike prior work that assumes independence, this work focuses on correlated distributions, reflecting practical constraints in domains such as reliability testing, drug discovery, and active search, where dependencies among variables are common.

The central challenge arises from two axes: (I) complexity of the target function $f$ under arbitrary correlations; (II) complexity of the correlation structure with fixed, tractable functions. Notably, SBFC encompasses and generalizes classical problems such as Min-Sum Set Cover ($1$-uniform matroid), $k$-Min Sum Set Cover ($\{X_e\}$0-uniform matroid), and partition matroid certification.

Main Results

Axis I: Structured Functions under Arbitrary Correlations

For certification functions corresponding to matroid spanning sets (matroid-SBFC), the paper establishes tight approximation guarantees:

• Non-adaptive SBFC for Matroids: An $\{X_e\}$1-approximation algorithm is proposed, provably tight up to constants (unless P = NP), even for partition matroid structures.
• Non-adaptive SBFC for $\{X_e\}$2-Uniform Matroids: For $\{X_e\}$3-of-$\{X_e\}$4 certification, a constant-factor $\{X_e\}$5-approximation is achieved, leveraging connections to Min Sum Set Cover and using advanced randomized rounding on matroid polytopes.

These results are based on linear programming relaxations augmented by knapsack cover inequalities and matroid constraints. The algorithms utilize convex reformulations and stochastic submodular function minimization, rendered tractable via sampling-based gradient estimation and polynomial-time stochastic optimization.

Axis II: Exploiting Correlation Structure

When the function is fixed (e.g., $\{X_e\}$6-of-$\{X_e\}$7), but correlations have further structure, markedly improved adaptive approximation is possible:

• Adaptive Graph Probing Problem: On graphs where edge variables inherit correlation from shared Bernoulli vertex variables, an adaptive $\{X_e\}$8-approximation algorithm is provided, substantially outperforming prior lower bounds on adaptivity gaps for SBFE with arbitrary correlations.
• Adaptive Hypergraph Probing: Generalizing to hypergraphs of rank $\{X_e\}$9, an adaptive $U$0-approximation is obtained, combining recursive probing strategies and independence recovery.
• Negative Dependence (CNA Distributions): For $U$1-of-$U$2 functions, the greedy probing order yields a $U$3-approximation under conditional negative association, improving over the previous tight $U$4-approximation for general read-once DNF.

A central technical result is the adaptivity gap analysis for Graph Probing, showing that adaptive algorithms can achieve polynomial improvements over non-adaptive strategies in correlated settings.

Algorithmic Techniques

The LP relaxations for matroid certification incorporate exponentially many knapsack cover-type constraints. Efficient separation is achieved by reducing constraint separation to parametric submodular minimization. Fractional solutions are rounded via matroid-based randomized rounding, achieving high expected rank coverage within the logarithmic or constant factor bounds.

For structured correlation models (graph/hypergraph probing), the recursive strategy preserves variable independence after each phase, ensuring that each adaptive phase only incurs linear or quadratic blowup in cost relative to the optimal. Adaptivity enables phase-wise reduction in the knapsack requirement, translating to logarithmic scaling in the approximation ratio.

The negative association analysis employs charging arguments based on marginal probabilities, mapping greedy probes to optimal probes and leveraging the conditional negative cylinder property to attain tighter bounds.

Implications and Future Directions

The results decisively delineate the outer limits of tractable SBFC certification under correlated uncertainty, establishing sharp algorithmic and hardness boundaries. Practically, these algorithms extend the applicability of stochastic function certification to systems with nontrivial dependencies, with immediate implications for sequential testing, active search, and reliability certification in complex settings.

Theoretically, the tight bounds and adaptivity analyses reveal a nuanced landscape: tractable certification is possible for well-structured functions (matroids, uniform and partition constraints) even under arbitrary correlations, but for functions with less structure, or sparse knowledge of the correlation model, adaptivity is mandatory and the gap is polynomial.

Future directions include extending these certification architectures to richer coverage functions (non-monotone, $U$5-of-$U$6, conjunctive/disjunctive normal forms), hybrid models with partial independence, and further exploration of negative dependence beyond CNA. Moreover, the stochastic convex optimization strategies may find broader use in correlated stochastic optimization, enriching the methodological toolkit for sequential decision-making under uncertainty in AI systems.

Conclusion

This work establishes a comprehensive theory for Stochastic Boolean Function Certification under correlated distributions, providing tight approximation algorithms for matroid-based functions, improved guarantees for $U$7-of-$U$8 and $U$9-of-$p$0 with negative dependence, and demonstrating the crucial impact of adaptivity for structured correlation models. The theoretical foundation and algorithmic machinery developed are poised to influence the design of certification protocols and sequential testing in stochastic systems with complex dependencies.