arXiv subjects

Lisa Hellerstein

Publications and source records attributed to Lisa Hellerstein.

22 records · Page 2Linked to original sources

arXiv2013

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.

cs.DS↗

arXiv2012

Max-Throughput for (Conservative) k-of-n Testing

We define a variant of k-of-n testing that we call conservative k-of-n testing. We present a polynomial-time, combinatorial algorithm for the problem of maximizing throughput of conservative k-of-n testing, in a parallel setting. This extends previous work of Kodialam and Condon et al., who presented combinatorial algorithms for parallel pipelined filter ordering, which is the special case where k=1 (or k = n). We also consider the problem of maximizing throughput for standard k-of-n testing, and show how to obtain a polynomial-time algorithm based on the ellipsoid method using previous techniques.

cs.DS↗

arXiv2011

Tight Bounds on Proper Equivalence Query Learning of DNF

We prove a new structural lemma for partial Boolean functions $f$, which we call the seed lemma for DNF. Using the lemma, we give the first subexponential algorithm for proper learning of DNF in Angluin's Equivalence Query (EQ) model. The algorithm has time and query complexity $2^{(\tilde{O}{\sqrt{n}})}$, which is optimal. We also give a new result on certificates for DNF-size, a simple algorithm for properly PAC-learning DNF, and new results on EQ-learning $\log n$-term DNF and decision trees.

cs.LG↗

arXiv2011

On the gap between ess(f) and cnf_size(f)

Given a Boolean function f, the quantity ess(f) denotes the largest set of assignments that falsify f, no two of which falsify a common implicate of f. Although ess(f)$ is clearly a lower bound on cnf_size(f) (the minimum number of clauses in a CNF formula for f), Cepek et al. showed that it is not, in general, a tight lower bound. They gave examples of functions f for which there is a small gap between ess(f) and cnf_size(f). We demonstrate significantly larger gaps. We show that the gap can be exponential in n for arbitrary Boolean functions, and Theta(sqrt{n}) for Horn functions, where n is the number of variables of f. We also introduce a natural extension of the quantity ess(f), which we call ess_k(f), which is the largest set of assignments, no k of which falsify a common implicate of f.

cs.DM↗