PROBABILISTIC ANALYSIS OF A BIN COVERING ALGORITHM

In the bin covering problem we are asked to pack a list X(n) = (x(1),x (2),...,x(n)) of n items, each with size no larger than one, into the maximum number of bins such that the sum of the sizes of the items in each bin is al least one. In this article we analyze the asymptotic av erage-case behavior of the Iterated-Lowest-Fit-Decreasing (ILFD) algor ithm proposed by Assmann et al. Let OPT(X(n)) denote the maximum numbe r of bins that can be covered by X(it) and let ILFD(X(n)) denote the n umber of bins covered by the ILFD algorithm. Assuming that X(iz) is a random sample from an arbitrary probability measure mu over [0, 1], we show the existence of a constant d(mu) and a constructible sequence { Xi(n) is an element of [0, 1](n): n greater than or equal to 1} such t hat /(ILFD(Xi(n))/n) - d(mu)/less than or equal to 1/n and lim(n-->inf inity) (ILFD(X(n))/n) = d(mu), almost surely. Since (ILFD(X(n))/n) alw ays lies in [0, 1], it follows that lim(n-->infinity) (E [ILFD(X(n))]/ n) = d(mu) as well. We also show that the expected values of the ratio r(ILFD)(X(n)) = OPT(X(n))/ILFD(X(n)), over all possible probability m easures for X(n), lie in [1, 4/3], the same range as the deterministic case.