Bin packing and covering problems with rejection

Abstract

In this paper we consider the following problems: We are given a set of n items {u1, ⋯, un}, each item ui is characterized by its size wi ∈ (0, 1] and its penalty/profit pi ≥ 0, and a number of unitcapacity bins. An item can be either rejected, in which case we pay/get its penalty/profit, or put into one bin under the constraint that the total size of the items in the bin is not greater/smaller than 1. No item can be spread into more than one bin. The objective is to minimize/maximize the sum of the number of used/covered bins and the penalties/profits of all rejected items. We call the problems bin packing/covering with rejection penalties/profits, and denoted by BPR and BCR respectively. For the online BPR problem, we present an algorithm with an absolute competitive ratio of 2.618 while the lower bound is 2.343, and an algorithm with an asymptotic competitive ratio of arbitrarily close to 7/4 while the lower bound is 1.540. For the offline BPR problem, we present an algorithm with an absolute worst-case ratio of 2 while the lower bound is 3/2, and an algorithm with an asymptotic worst-case ratio of 3/2. For the online BCR problem, we show that no algorithm can have an absolute competitive ratio of greater than 0, and present an algorithm with an asymptotic competitive ratio of 1/2, which is the best possible. For the offline BCR problem, we also present an algorithm with an absolute worst-case ratio of 1/2 which matches the lower bound. © Springer-Verlag Berlin Heidelberg 2005.