An Efficient Approximation Scheme for Variable-Sized Bin Packing

Abstract

In the classical bin packing problem one is required to pack a given list of items into the smallest possible number of unit-sized bins. Because this problem is NP-complete, researchers have tried to find efficient approximation algorithms that solve this problem in a reasonable amount of time. If we let $A(I)$ be the number of bins used by algorithm A to pack a list of items I, and ${\textit{OPT}}(I)$ be the minimum number of bins necessary to pack I, one defines the asymptotic performance ratio of A to be ${{A(I)} / {{\textit{OPT}}(I)}}$, as ${\textit{OPT}}(I)$ tends to infinity. De la Vega and Lueker presented an approximation scheme, which for any $ \in > 0$, yields an approximation algorithm with performance ratio $1 + \varepsilon $. This scheme has time complexity polynomial in n, the number of items, but exponential in ${1 / \varepsilon }$. Karmarkar and Karp extended this scheme to one that has time complexity polynomial in n and ${1 / \varepsilon }$. In the variable-sized bin packing problem, one...