Non-Adaptive Evaluation of $k$-of-$n$ Functions: Tight Gap and a Unit-Cost PTAS

2025-07-08

AI Summary

This paper studies the optimal evaluation of $k$-of-$n$ Boolean functions under non-adaptive querying, where the goal is to determine whether at least $k$ out of $n$ variables equal 1 while minimizing the expected testing cost. Addressing the lack of efficient algorithms for this Sequential Boolean Function Evaluation (SBFE) problem in the non-adaptive setting, we establish a tight lower bound of 2 on the adaptivity gap—the ratio between optimal non-adaptive and adaptive costs. For unit testing costs, we present the first polynomial-time approximation scheme (PTAS), introducing two key conceptual innovations: “bidirectional dominance” and “milestone tests.” Our algorithm employs a divide-and-conquer framework, integrating randomized shifting and a two-stage dominance analysis to achieve, in polynomial time, an arbitrarily close approximation to the optimal non-adaptive strategy. This work delivers the first PTAS for any non-adaptive SBFE problem, significantly advancing the theoretical understanding of adaptivity gaps and expanding the frontier of algorithm design for non-adaptive Boolean function evaluation.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We consider the Stochastic Boolean Function Evaluation (SBFE) problem in the well-studied case of $k$-of-$n$ functions: There are independent Boolean random variables $x_1,dots,x_n$ where each variable $i$ has a known probability $p_i$ of taking value $1$, and a known cost $c_i$ that can be paid to find out its value. The value of the function is $1$ iff there are at least $k$ $1$s among the variables. The goal is to efficiently compute a strategy that, at minimum expected cost, tests the variables until the function value is determined. While an elegant polynomial-time exact algorithm is known when tests can be made adaptively, we focus on the non-adaptive variant, for which much less is known. First, we show a clean and tight lower bound of $2$ on the adaptivity gap, i.e., the worst-case multiplicative loss in the objective function caused by disallowing adaptivity, of the problem. This improves the tight lower bound of $3/2$ for the unit-cost variant. Second, we give a PTAS for computing the best non-adaptive strategy in the unit-cost case, the first PTAS for an SBFE problem. At the core, our scheme establishes a novel notion of two-sided dominance (w.r.t. the optimal solution) by guessing so-called milestone tests for a set of carefully chosen buckets of tests. To turn this technique into a polynomial-time algorithm, we use a decomposition approach paired with a random-shift argument.

Problem

• Evaluates non-adaptive strategies for k-of-n Boolean functions
• Proves tight adaptivity gap of 2 for non-adaptive evaluation
• Provides first PTAS for unit-cost non-adaptive SBFE problem

Innovation

• Non-adaptive strategy with tight adaptivity gap
• PTAS for unit-cost non-adaptive evaluation
• Two-sided dominance via milestone tests