We present a new approximation algorithm for the stochastic submodular set cover (SSSC) problem called adaptive dual greedy. We use this algorithm to obtain a 3-approximation algorithm solving the stochastic Boolean function evaluation (SBFE) problem for linear threshold formulas (LTFs). We also obtain a 3approximation algorithm for the closely related stochastic min-knapsack problem and a 2-approximation for a variant of that problem. We prove a new approximation bound for a previous algorithm for the SSSC problem, the adaptive greedy algorithm of Golovin and Krause. We also consider an approach to approximating SBFE problems using the adaptive greedy algorithm, which we call the Q-value approach. This approach easily yields a new result for evaluation of CDNF (conjunctive / disjunctive normal form) formulas, and we apply variants of it to simultaneous evaluation problems and a ranking problem. However, we show that the Q-value approach provably cannot be used to obtain a sublinear approximation factor for the SBFE problem for LTFs or read-once disjunctive normal form formulas.
Many problems in Machine Learning can be modeled as submodular optimization problems. Recent work has focused on stochastic or adaptive versions of these problems. We consider the Scenario Submodular Cover problem, which is a counterpart to the Stochastic Submodular Cover problem studied by Golovin and Krause (2011). In Scenario Submodular Cover, the goal is to produce a cover with minimum expected cost, where the expectation is with respect to an empirical joint distribution, given as input by a weighted sample of realizations. In contrast, in Stochastic Submodular Cover, the variables of the input distribution are assumed to be independent, and the distribution of each variable is given as input. Building on algorithms developed by Cicalese et al. (2014) and Golovin and Krause (2011) for related problems, we give two approximation algorithms for Scenario Submodular Cover over discrete distributions. The first achieves an approximation factor of O(log Qm), where m is the size of the sample and Q is the goal utility. The second, simpler algorithm achieves an approximation bound of O(log QW ), where Q is the goal utility and W is the sum of the integer weights. (Both bounds assume an integer-valued utility function.) Our results yield approximation bounds for other problems involving non-independent distributions that are explicitly specified by their support.
Stochastic boolean function evaluation (SBFE) is the problem of determining the value of a given boolean function f on an unknown input x, when each bit x i of x can only be determined by paying a given associated cost c i . Further, x is drawn from a given product distribution: for each x i , Pr[x i = 1] = p i and the bits are independent. The goal is to minimize the expected cost of evaluation. In this paper, we study the complexity of the SBFE problem for classes of DNF formulas. We consider both exact and approximate versions of the problem for subclasses of DNF, for arbitrary costs and product distributions, and for unit costs and/or the uniform distribution.
Deshpande et al. presented a k(ln R + 1) approximation bound for Stochastic Submodular Cover, where k is the state set size, R is the maximum utility of a single item, and the utility function is integer-valued. This bound is similar to the ln Q/(eta+1) bound given by Golovin and Krause, whose analysis was recently found to have an error. Here Q >= R is the goal utility and eta is the minimum gap between Q and any attainable utility Q' < Q. We revisit the proof of the k(ln R + 1) bound of Deshpande et al., fill in the details of the proof of a key lemma, and prove two bounds for real-valued utility functions: k(ln R_1 + 1) and (ln R_E + 1). Here R_1 equals the maximum ratio between the largest increase in utility attainable from a single item, and the smallest non-zero increase attainable from that same item (in the same state). The quantity R_E equals the maximum ratio between the largest expected increase in utility from a single item, and the smallest non-zero expected increase in utility from that same item. Our bounds apply only to the stochastic setting with independent states.
We consider the problem of stochastic monotone submodular function maximization, subject to constraints. We give results on adaptivity gaps, and on the gap between the optimal offline and online solutions. We present a procedure that transforms a decision tree (adaptive algorithm) into a non-adaptive chain. We prove that this chain achieves at least τ times the utility of the decision tree, over a product distribution and binary state space, where τ = min i,j Pr[x i = j]. This proves an adaptivity gap of 1 τ (which is 2 in the case of a uniform distribution) for the problem of stochastic monotone submodular maximization subject to state-independent constraints. For a cardinality constraint, we prove that a simple adaptive greedy algorithm achieves an approximation factor of (1 − 1 e τ ) with respect to the optimal offline solution; previously, it has been proven that the algorithm achieves an approximation factor of (1 − 1 e ) with respect to the optimal adaptive online solution. Finally, we show that there exists a non-adaptive solution for the stochastic max coverage problem that is within a factor (1 − 1 e ) of the optimal adaptive solution and within a factor of τ(1 − 1 e ) of the optimal offline solution.
Recently, Deshpande et al. introduced a new measure of the complexity of a Boolean function. We call this measure the "goal value" of the function. The goal value of f is defined in terms of a monotone, submodular utility function associated with f . As shown by Deshpande et al., proving that a Boolean function f has small goal value can lead to a good approximation algorithm for the Stochastic Boolean Function Evaluation problem for f . Also, if f has small goal value, it indicates a close relationship between two other measures of the complexity of f , its average-case decision tree complexity and its average-case certificate complexity. In this paper, we explore the goal value measure in detail. We present bounds on the goal values of arbitrary and specific Boolean functions, and present results on properties of the measure. We compare the goal value measure to other, previously studied, measures of the complexity of Boolean functions. Finally, we discuss a number of open questions provoked by our work.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.