Lecture 4 : More Measure Concentration : Counting DNF Satisfying Assignments
نویسنده
چکیده
Suppose there is a bag containing red balls and blue balls. You would like to estimate the fraction of red balls in the bag. However, you are only allowed to sample randomly from the bag with replacement. The straightforward method is to make T random samples and use the fraction of red balls in the random samples as an estimate of the true fraction. The question is: how many samples are enough so that with probability at least 1− , our estimate is within a multiplicative error of 0 < < 1 from the true value?
منابع مشابه
Lecture 25
Given a DNF formula φ with n variables, DNF counting is the problem finding the number of satisfying assignments for φ. Note that in general satisfiability for DNF is easy as we need only satisfy a single clause, but the counting problem is hard. Indeed, if we could do this, then given any 3-CNF formula f with n variables, we could take its negation, count how many satisfying assignments its ne...
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We develop polynomial time Monte-Carlo algorithms which produce good approximate solutions to enumeration problems for which it is known that the computation of the exact solution is very hard. We start by developing a Monte-Carlo approximation algorithm for the DNF counting problem, which is the problem of counting the number of satisfying truth assignments to a formula in disjunctive normal f...
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We describe a deterministic algorithm that, for constant k, given a k-DNF or kCNF formula φ and a parameter ε, runs in time linear in the size of φ and polynomial in 1/ε and returns an estimate of the fraction of satisfying assignments for φ up to an additive error ε. For k-DNF, a multiplicative approximation is also achievable in time polynomial in 1/ε and linear in the size of φ. Previous alg...
متن کامل.1 Counting Dnf Satisfying Assignments
Suppose there is a bag containing red balls and blue balls. You would like to estimate the fraction of red balls in the bag. However, you are only allowed to sample randomly from the bag with replacement. The straightforward method is to make T random samples and use the fraction of red balls in the random samples as an estimate of the true fraction. The question is: how many samples are enough...
متن کاملRandomized Algorithms 2013 A Lecture 11 – Importance Sampling , and the Local Lemma ∗
Problem definition: The input is a DNF formula f with m clauses C1, . . . , Cm over n variables x1, . . . , xn, i.e. f = ∨i=1Ci where each Ci is the conjunction of literals like x2 ∧ x̄5 ∧ xn. The goal is the estimate the number of Boolean assignments that satisfy f . Theorem 1 [Karp and Luby, 1983]: Let S ⊂ {0, 1}n be the set of satisfying assignments. Then |S| can be estimated within factor 1 ...
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