Polynomiality for Bin Packing with a Constant Number of Item Types

TL;DR

This paper proves that bin packing with a fixed number of item types can be solved in polynomial time, resolving an open problem for all cases where the number of item types is three or more.

Contribution

It introduces a polynomial-time algorithm for bin packing with a constant number of item types, extending to related high multiplicity scheduling problems.

Findings

Polynomial-time algorithm for fixed number of item types in bin packing.

Extension of the approach to high multiplicity scheduling problems.

Resolution of an open problem for d >= 3 in bin packing.

Abstract

We consider the bin packing problem with d different item sizes s_i and item multiplicities a_i, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem. In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d >= 3. In fact, for constant d our algorithm solves the following problem in polynomial time: given two d-dimensional polytopes P and Q, find the smallest number of integer points in P whose sum lies in Q. Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.