Introduction

Abstract

The Monte Carlo method provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments on a computer. Remarkably, the method applies to problems with absolutely no probabilistic content as well as to those with inherent probabilistic structure. This alone does not give the Monte Carlo method an advantage over other methods of approximation. However, among all numerical methods that rely on n-point evaluations in m-dimensional space to produce an approximate solution, the Monte Carlo method has absolute error of estimate that decreases as n ^−l/2 whereas, in the absence of exploitable special structure, all others have errors that decrease as n ^−l/m at best. This property gives the Monte Carlo method a considerable edge in computational efficiency as m, the size of the problem, increases. Combinatorial settings illustrate this property especially well. Whereas the exact solution to a combinatorial problem with m elements often has computational cost that increases exponentially or superexponentially with m, the Monte Carlo method. frequently provides an estimated solution with tolerable error at a cost that increases no faster than as a polynomial in m.