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

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.