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

TL;DR

A PTAS is given for computing the best non-adaptive strategy in the unit-cost case, the first PTAS for an SBFE problem and establishes a novel notion of two-sided dominance by guessing so-called milestone tests for a set of carefully chosen buckets of tests.

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. In fact, our PTAS extends to the class of arbitrary symmetric Boolean functions, which are Boolean functions whose value only depends on the number of $1$s among the input variables.

Figures (2)

• Figure 1: Example of output generated by \ref{['alg:alg']} for a single iteration of the main loop.
• Figure 2: Illustration of the setup in the proof of \ref{['claim:alg']}. Shaded boxes indicate elements in the corresponding set.

Theorems & Definitions (30)

• Theorem 1
• Theorem 2
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of \ref{['thm:gapmain']}
• Claim 5
• proof : Proof of \ref{['claim:nonadaptivestop']}