Bin packing under linear constraints

Abstract

In this paper, we study a bin packing problem in which the sizes of items are determined by k linear constraints, and the goal is to decide the sizes of items and pack them into a minimal number of unit sized bins. We first provide two scenarios that motivate this research. We show that this problem is NP-hard in general, and propose several algorithms in terms of the number of constraints. If the number of constraints is arbitrary, we propose a 2-approximation algorithm, which is based on the analysis of the Next Fit algorithm for the bin packing problem. If the number of linear constraints is a fixed constant, then we obtain a PTAS by combining linear programming and enumeration techniques, and show that it is an optimal algorithm in polynomial time when the number of constraints is one or two. It is well known that the bin packing problem is strongly NP-hard and cannot be approximated within a factor 3 / 2 unless P = NP. This result implies that the bin packing problem with a fixed number of constraints may be simper than the original bin packing problem. Finally, we discuss the case when the sizes of items are bounded.