[Paper Review] About the structure of the integer cone and its application to bin packing

Klaus Jansen, Kim-Manuel Klein|arXiv (Cornell University)|Jan 16, 2017

Optimization and Packing Problems6 citations

TL;DR

This paper presents a new structure theorem for the integer cone in bin packing by leveraging the vertices of the underlying integer knapsack polytope, leading to a fixed-parameter tractable algorithm with running time |V|²ᴼ⁽ᵈ⁾ · enc(I)ᴼ⁽¹⁾, where |V| is the number of vertices and d is the number of item sizes. The result shows efficient solvability when the polytope has few vertices, and the bounds are proven asymptotically tight via number-theoretic constructions.

ABSTRACT

We consider the bin packing problem with d different item sizes and revisit the structure theorem given by Goemans and Rothvos [5] 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|2O(d) · 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.

Motivation & Objective

• To re-express the structure of solutions in the integer cone of bin packing using the vertices of the integer knapsack polytope.
• To develop a new algorithmic framework that achieves fixed-parameter tractability in the number of vertices of the knapsack polytope.
• To establish tight asymptotic bounds for the structural theorem by constructing bin packing instances using number-theoretic principles.
• To demonstrate that bin packing becomes efficiently solvable when the underlying integer knapsack polytope has a simple structure with few vertices.

Proposed method

• Reformulate the solution structure of the integer cone by focusing on the vertex set V of the integer knapsack polytope instead of other combinatorial constructs.
• Introduce novel techniques to modify integer cone solutions while preserving feasibility and optimality, enabling structural analysis.
• Design an algorithm that runs in time |V|²ᴼ⁽ᵈ⁾ · enc(I)ᴼ⁽¹⁾, making it fixed-parameter tractable with respect to |V|.
• Use classical number-theoretic theorems to construct bin packing instances that achieve the lower bound on structural complexity.
• Prove the asymptotic tightness of the bounds by showing that the constructed instances match the theoretical limits of the structure theorem.

Experimental results

Research questions

• RQ1 Can the structure of the integer cone in bin packing be redefined using the vertices of the underlying integer knapsack polytope?
• RQ2 What is the computational complexity of bin packing when parameterized by the number of vertices of the knapsack polytope?
• RQ3 How tight are the bounds of the new structure theorem in terms of worst-case instance construction?
• RQ4 Can number-theoretic constructions be used to generate instances that achieve the theoretical lower bound on structural complexity?
• RQ5 Is the proposed algorithm efficient when the integer knapsack polytope has few vertices?

Key findings

• The paper establishes a new structure theorem for the integer cone that relies on the vertex set V of the integer knapsack polytope, offering a more refined characterization than prior work.
• An algorithm is developed with running time |V|²ᴼ⁽ᵈ⁾ · enc(I)ᴼ⁽¹⁾, proving fixed-parameter tractability when parameterized by |V|.
• The bounds of the structure theorem are asymptotically tight, as demonstrated by constructing instances that meet the theoretical limits.
• The construction of these tight instances relies on deep insights from classical number theory, linking combinatorial optimization with number-theoretic principles.
• The results show that bin packing is efficiently solvable when the underlying integer knapsack polytope has a simple structure with few vertices.