KARPLUBY - M cnte-earlo AlgOrithIIlS for Enumeration and...

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Mcnte-earlo AlgOrithIIlS for Enumeration and Reliability Problems Richard M. Karpt University oJ California at Berkeley Michael Lubyt 01 Toronto 1. Introduction We present a simple but very general Monte-Carlo technique for the approximate solution of enumeration and reliability prob- lems. Several applications are given, includ- ing: 1. Estimating the number of triangulated plane maps with a given number of ver- tices; 2. Estimating the cardinality of a union of sets; 3. Estimating the number of input combi- nations for which a boolean function, presented in disjunctive normal form, assume the value true; 4. Estimating the failure probability of a system with faulty components. 1.1 Randomized Approximation Algorithms and Approximation Schemes Let f be a function ~rom some domain D into the positive reals. shall be con- cerned with randomized algorithms which accept as input any wED and produce as output a positive real number 1 (w) which is an estimate of f (w). Since the algorithm involves randomization, 1 is a random variable, rather than a constant, for each :fixed w. Such a randomized algorithm is called an (l:,o) approximation algorithm for J if, for every input wED, Pr [I l( w )- t {w } I > l; 1 < 0 J{w) . tResearch supported by NSF Grant MCS-81-05217 0272-5428/83/0000/0056$01.00 © 1983 IEEE 56 In a similar spirit, we can discuss random- ized approximation methods in which ~ 0, as'well as w, are part of the input. Such a randomized algorithm is called a random- ized approximation scheme for f if, every input triple (l:,O,w), where E > 0 0 < 0 < 1, the algorithm produces as output a real number l£,o(w) such that ['le'~jl~f(W) I > e I < l5 In cases where the domain D is a set of strings, a randomized approximation scheme is called fully polynomial if its exe- cution time is bounded a polynomial in ;. ~ and the length of We derive ran- domized schemes the problems mentioned above. In particular, we give a polynomial scheme estimating the number of input combina- tions that make a disjunctive normal form boolean formula true. Thus we have a scheme for a IP complete problem. 2. Counting Equivalence Classes The general principles underlying all our results can be described abstractly as fol- lows. S be a finite set on which an equivalence relation '"'J is defined. We wish to estimate the number of equivalence classes into which I'V partitions S. The number of equivalence classes will be denoted 1 S /I'J I. We assume that 1 s I J the cardinality of S, is known. [x] denote the equivalence class containing % . two Monte-Carlo methods for estimating'S /f'V ,. Each of these methods
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executes t trials. The estimator of IS/row I is X +X + ... +X 1 2 t ,where Xi is the result of t the it", trial. The random variables ~ are independent and identically distributed, and E [~] = I S /~ Each of the methods requires a pro- cedure for choosing elements at random from S. Method 1 assumes that, given %, we can determine the number of elements in [x ].
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KARPLUBY - M cnte-earlo AlgOrithIIlS for Enumeration and...

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