Average-case analyses of first fit and random fit bin packing

Abstract

ABSTRACT:We prove that the First Fit bin packing algorithm is stable under the input distribution U k − 2 k for all k ≥ 3, settling an open question from the recent survey by Coffman, Garey, and Johnson ["Approximation algorithms for bin backing: A survey," Approximation algorithms for NP-hard problems, D. Hochbaum (Editor), PWS, Boston, 1996]. Our proof generalizes the multidimensional Markov chain analysis used by Kenyon, Sinclair, and Rabani to prove that Best Fit is also stable under these distributions [Proc Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 1995, pp. 351-358]. Our proof is motivated by an analysis of Random Fit, a new simple packing algorithm related to First Fit, that is interesting in its own right. We show that Random Fit is stable under the input distributions U k − 2 k , as well as present worst case bounds and some results on distributions U k − 1 k and U k k for Random Fit.

Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing

“…Some of this has been proved. In [19] it was shown that EW BF n (U {k − 2, k}) = O(1) for all k > 0, and this result was extended to F F in [1]. In [4] it was shown that…”

“…The expected waste rate EW A n (F ) for an algorithm A and distribution F is defined to be the expected value of W A (L n (F )) as a function of n. In what follows we typically abbreviate this as simply the "expected waste." We say a distribution F is a bounded waste distribution if EW OP T n (F ) = O (1). As a consequence of Theorem 1 and a classification theorem of Courcoubetis and Weber [11], we can prove the following.…”

On the sum-of-squares algorithm for bin packing

“…For the on-line algorithms FF and BF, the situation is no better. Although they can be shown to have O(1) waste when j = O(v/k) [2], when j = k -2 [1,12], and (in the case of BF) for specified pairs (j, k) with k _< 14 [5], for most values of (j, k) it appears experimentally that the expected waste of BF and FF is linear, and this has been proved for BF and the pairs (8,11) and (9,12) Turning to less distribution-specific results, the first relevant results concerned off-line algorithms. Karmarkar and Karp in [11] presented an off-fine deterministic polynomial time algorithm KI( that for any list L in our setting guarantees that KK(L) < OPT(L)+O(log 2 B).…”

“…The average case behavior under discrete distributions for standard heuristics has been studied in [1,2,3,5,6,12]. These papers concentrated on the discrete uniform distributions U{j, k} mentioned above, where the bin capacity B = k and the item sizes are 1,2,... ,j < k, all equally likely.…”

Online Stochastic Bin Packing

“…, J B , Coffman et al [5] showed the expected waste for J = B or J = B − 1 grows as Θ(nB 1 2 ) for First Fit, and Θ(n 1 2 log B) for Best Fit. For J = B − 2, bounded expected waste for Best Fit was showed by Kenyon et al [11], and for First Fit (using Random Fit as an intermediate step) by Albers and Mitzenmacher [1]. Kenyon and Mitzenmacher [10] proved that the waste under Best Fit is linear when J = αB, and 99 150 < α < 100 150 , B large enough but is conjectured to hold for all 0 < α < 1.…”