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 Ufk \Gamma 2; kg for all k 3, settling an open question from the recent survey by Coffman, Garey, and Johnson [3]. Our proof generalizes the multi-dimensional Markov chain analysis used by Kenyon, Rabani, and Sinclair to prove that Best Fit is also stable under these distributions [10]. 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 Ufk \Gamma 2; kg, as well as present worst-case bounds and some results on distributions Ufk \Gamma 1; kg and Ufk; kg for Random Fit. 1 Introduction In the one-dimensional bin packing problem, one is given a sequence a 1 ; : : : ; a n 2 (0; 1] of items to pack into bins of unit capacity so as to minimize the number of bins used. A great deal of literature has focused on this problem, perhaps because, as Coffm..