Approximation Algorithms for Variable-Sized and Generalized Bin Covering

In this paper, we consider the Generalized Bin Covering problem: We are given m bin types, where each bin of type i has profit pi and demand di. Furthermore, there are n items, where item j has size sj . A bin of type i is covered if the set of items assigned to it has total size at least the demand di. In that case, the profit of pi is earned and the objective is to maximize the total profit. To the best of our knowledge, only the cases pi = di = 1 (Bin Covering) and pi = di (Variable-Sized Bin Covering) have been treated before. We study two models of bin supply: In the unit supply model, we have exactly one bin of each type, i. e., we have individual bins. By contrast, in the infinite supply model, we have arbitrarily many bins of each type. Clearly, the unit supply model is a generalization of the infinite supply model, since we can simulate the latter with the former by introducing sufficiently many copies of each bin. To the best of our knowledge the unit supply model has not been studied yet. It is well-known that the problem in the infinite supply model is NP-hard, which can be seen by a straightforward reduction from Partition, and this hardness carries over to the unit supply model. This also implies that the problem can not be approximated better than two, unless P = NP. We begin our study with the unit supply model. Our results therein hold not only asymptotically, but for all instances. This contrasts most of the previous work on Bin Covering, which has been asymptotic. We prove that there is a combinatorial 5-approximation algorithm for Generalized Bin Covering with unit supply, which has running time O(nm√m + n). This also transfers to the infinite supply model by the above argument. Furthermore, for Variable-Sized Bin Covering, in which we have pi = di, we show that the natural and fast Next Fit Decreasing (nfd) algorithm is a 9/4-approximation in the unit supply model. The bound is tight for the algorithm and close to being best-possible, since the problem is inapproximable up to a factor of two, unless P = NP. Our analysis gives detailed insight into the limited extent to which the optimum can significantly outperform nfd. Then the question arises if we can improve on those results in asymptotic notions, where the optimal profit diverges. We discuss the difficulty of defining asymptotics in the unit supply model. For two natural definitions, the negative result holds that VariableSized Bin Covering in the unit supply model does not allow an APTAS. Clearly, this also excludes an APTAS for Generalized Bin Covering in that model. Nonetheless, we show that there is an AFPTAS for Variable-Sized Bin Covering in the infinite supply model. ∗Humboldt University of Berlin, Germany, [email protected] University of Freiburg, Germany, [email protected]