Anomalous behavior in bin packing algorithms

Abstract

In the classical bin packing problem one is given a list of items and asked to pack them into the fewest possible unit-sized bins. Given two lists, L 1 and L 2, where L 2 is derived from L 1 by deleting some elements of L 1 and/or reducing the size of some elements of L 1, one might hope that an approximation algorithm would use no more bins to pack L 2 than it uses to pack L 1. Johnson and Graham have given examples showing that First-Fit and First-Fit Decreasing can actually use more bins to pack L 2 than L 1. Graham has also studied this type of behavior among multiprocessor scheduling algorithms. In the present paper we extend this study of anomalous behavior to a broad class of approximation algorithms for bin packing. To do this we introduce a technique which allows one to characterize the monotonic/anomalous behavior of any algorithm in a large, natural class. We then derive upper and lower bounds on the anomalous behavior of the algorithms which are anomalous and provide conditions under which a normally nonmonotonic algorithm becomes monotonic.