The Support of Bin Packing is Exponential

TL;DR

This paper proves an exponential lower bound on the support size of solutions to the classical Bin Packing problem, matching known upper bounds and impacting algorithm complexity analysis.

Contribution

It establishes a tight exponential lower bound on the support of Bin Packing solutions and introduces a novel ILP aggregation technique with potential broader applications.

Findings

Lower bound on support is 2^{Ω(d)} matching upper bounds.

Implications for the time complexity of several Bin Packing algorithms.

Develops a new ILP aggregation technique that incorporates variable bounds.

Abstract

Consider the classical Bin Packing problem with $d$ different item sizes $s_i$ and amounts of items $a_i.$ The support of a Bin Packing solution is the number of differently filled bins. In this work, we show that the lower bound on the support of this problem is $2^{\Omega(d)}$. Our lower bound matches the upper bound of $2^d$ given by Eisenbrand and Shmonin [Oper.Research Letters '06] up to a constant factor. This result has direct implications for the time complexity of several Bin Packing algorithms, such as Goemans and Rothvoss [SODA '14], Jansen and Klein [SODA '17] and Jansen and Solis-Oba [IPCO '10]. To achieve our main result, we develop a technique to aggregate equality constrained ILPs with many constraints into an equivalent ILP with one constraint. Our technique contrasts existing aggregation techniques as we manage to integrate upper bounds on variables into the resulting constraint. We believe this technique can be useful for solving general ILPs or the $d$-dimensional knapsack problem.