About the Structure of the Integer Cone and its Application to Bin Packing

TL;DR

This paper introduces a new structure theorem for solutions to the integer cone in bin packing, leading to a fixed parameter tractable algorithm based on the vertices of the underlying integer polytope, with tight bounds.

Contribution

It presents a novel structure theorem for the integer cone in bin packing and an efficient algorithm parameterized by the number of vertices of the integer polytope.

Findings

The new structure theorem relies on the vertices of the integer polytope.

The algorithm runs in fixed parameter tractable time based on vertex count.

Bounds of the structure theorem are shown to be asymptotically tight.

Abstract

We consider the bin packing problem with $d$ different item sizes and revisit the structure theorem given by Goemans and Rothvo\ss [6] about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new structure theorem, we obtain an algorithm for the bin packing problem with running time $|V|^{2^{O(d)}} \cdot enc(I)^{O(1)}$, where $V$ is the set of vertices of the integer knapsack polytope and $enc(I)$ is the encoding length of the bin packing instance. The algorithm is fixed parameter tractable, parameterized by the number of vertices of the integer knapsack polytope $|V|$. This shows that the bin packing problem can be solved efficiently when the underlying integer knapsack polytope has an easy structure, i.e. has a small number of vertices. Furthermore, we show that the presented bounds of the structure theorem are asymptotically tight. We give a construction of bin packing instances using new structural insights and classical number theoretical theorems which yield the desired lower bound.