Probabilistic analysis of a bin covering algorithm

In the bin covering problem we are asked to pack a list X(n) = (x1, x2, . ,x,) 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 at least one. In this article we analyze the asymptotic average-case behavior of the Iterated-Lowest-Fit-Decreasing (IUD) algorithm proposed by Assmann et al. Let OPT(X(n)) denote the maximum number of bins that can be covered by X(n) and let ZLFD(X(n)) denote the number of bins covered by the ILFD algorithm. Assuming that X(n) is a random sample from an arbitrary probability measure p over [0, 11, we show the existence of a constant d(p) and a constructible sequence {Z(n) E [0, 11”: n > 11 such that I(ILFD(3(n))/n) d(p)1 < l/n and lim,,, (ILFD(X(n))/n) = d(p), almost surely. Since (ILFD(X(n))/n) always lies in [0, 11, it follows that lim,,, (E[ILFD(X(n))]/n) = d(p) as well. We also show that the expected values of the ratio r,,,,(X(n)) = OPT(X(n))/ILFD(X(n)), over all possible probability measures for X(n), lie in [l, $1, the same range as the deterministic case. Kqvwords: Bin covering; Bin packing; NP-hard; Heuristic; Average-case analysis