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Any randomized algorithm may be interpreted as a randomized choice among deterministic algorithms, and thus as a mixed strategy for Alice. Similarly, a non-random algorithm may be thought of as a pure strategy for Alice. In any two-player zero-sum game, if one player chooses a mixed strategy, then the other player has an optimal pure strategy ...
Seidel (1991) gave an algorithm for low-dimensional linear programming that may be adapted to the LP-type problem framework. Seidel's algorithm takes as input the set S and a separate set X (initially empty) of elements known to belong to the optimal basis. It then considers the remaining elements one-by-one in a random order, performing ...
The deterministic algorithm emulates the randomized rounding scheme: it considers each set in turn, and chooses ′ {,}. But instead of making each choice randomly based on x ∗ {\displaystyle x^{*}} , it makes the choice deterministically , so as to keep the conditional probability of failure, given the choices so far, below 1 .
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are ...
Indeed, this randomization principle is known to be a simple and effective way to obtain algorithms with almost certain good performance uniformly across many data sets, for many sorts of problems. Stochastic optimization methods of this kind include:
There are three optimal set covers, each of which includes two of the three given sets. Thus, the optimal value of the objective function of the corresponding 0–1 integer program is 2, the number of sets in the optimal covers. However, there is a fractional solution in which each set is assigned the weight 1/2, and for which the total value ...
Some problems which do not have a PTAS may admit a randomized algorithm with similar properties, a polynomial-time randomized approximation scheme or PRAS.A PRAS is an algorithm which takes an instance of an optimization or counting problem and a parameter ε > 0 and, in polynomial time, produces a solution that has a high probability of being within a factor ε of optimal.
In computing, a Monte Carlo algorithm is a randomized algorithm whose output may be incorrect with a certain (typically small) probability. Two examples of such algorithms are the Karger–Stein algorithm [ 1 ] and the Monte Carlo algorithm for minimum feedback arc set .