Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover

Stochastic Boolean Function Evaluation is the problem of determining the value of a given Boolean function f on an unknown input x, when each bit of x_i of x can only be determined by paying an associated cost c_i. The assumption is that x is drawn from a given product distribution, and the goal is to minimize the expected cost. This problem has been studied in Operations Research, where it is known as "sequential testing" of Boolean functions. It has also been studied in learning theory in the context of learning with attribute costs. We consider the general problem of developing approximation algorithms for Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for evaluating Boolean linear threshold formulas. We also present an approximation algorithm for evaluating CDNF formulas (and decision trees) achieving a factor of O(log kd), where k is the number of terms in the DNF formula, and d is the number of clauses in the CNF formula. In addition, we present approximation algorithms for simultaneous evaluation of linear threshold functions, and for ranking of linear functions. Our function evaluation algorithms are based on reductions to the Stochastic Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and Krause. They presented an approximation algorithm for the problem, called Adaptive Greedy. Our main technical contribution is a new approximation algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito, which is a generalization of Hochbaum's algorithm for the classical Set Cover Problem. We also give a new bound on the approximation achieved by the Adaptive Greedy algorithm of Golovin and Krause

The Stochastic Score Classification Problem

Consider the following Stochastic Score Classification Problem. A doctor is assessing a patient\u27s risk of developing a certain disease, and can perform n tests on the patient. Each test has a binary outcome, positive or negative. A positive result is an indication of risk, and a patient\u27s score is the total number of positive test results. Test results are accurate. The doctor needs to classify the patient into one of B risk classes, depending on the score (e.g., LOW, MEDIUM, and HIGH risk). Each of these classes corresponds to a contiguous range of scores. Test i has probability p_i of being positive, and it costs c_i to perform. To reduce costs, instead of performing all tests, the doctor will perform them sequentially and stop testing when it is possible to determine the patient\u27s risk category. The problem is to determine the order in which the doctor should perform the tests, so as to minimize expected testing cost. We provide approximation algorithms for adaptive and non-adaptive versions of this problem, and pose a number of open questions

(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing

Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p_e. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size. The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive - decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values. Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between O(k log k) and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [Gupta et al., 2017]