A Tight Double-Exponentially Lower Bound for High-Multiplicity Bin Packing
Klaus Jansen, Felix Ohnesorge, Lis Pirotton
TL;DR
The paper proves an ETH-based lower bound for high-multiplicity Bin Packing parameterized by the number of item sizes d, showing there is no algorithm running in time $(|I|^2)^{o(d)}$ unless ETH fails. It achieves this via a novel reduction from 3-SAT to an efficiently encodable ILP with $O(\log n)$ variables, which is then transformed into Bin Packing instances with $d=O(\log n)$. The construction uses a base-$\gamma$ encoding and ILP-aggregation to a single knapsack constraint, ensuring a tight correspondence between SAT satisfiability and bin-packing feasibility within a fixed bound. This establishes that the Goemans–Rothvoss doubly exponential runtime in d is optimal under ETH and opens avenues for applying compact ILP encodings to other high-mimension/low-variable problems.
Abstract
Consider a high-multiplicity Bin Packing instance $I$ with $d$ distinct item types. In 2014, Goemans and Rothvoss gave an algorithm with runtime ${{|I|}^2}^{O(d)}$ for this problem~[SODA'14], where $|I|$ denotes the encoding length of the instance $I$. Although, Jansen and Klein~[SODA'17] later developed an algorithm that improves upon this runtime in a special case, it has remained a major open problem by Goemans and Rothvoss~[J.ACM'20] whether the doubly exponential dependency on $d$ is necessary. We solve this open problem by showing that unless the ETH fails, there is no algorithm solving the high-multiplicity Bin Packing problem in time ${{|I|}^2}^{o(d)}$. To prove this, we introduce a novel reduction from 3-SAT. The core of our construction is efficiently encoding the entire information from a 3-SAT instance with $n$ variables into an ILP with $O(\log(n))$ variables. This result confirms that the Goemans and Rothvoss algorithm is best-possible for Bin Packing parameterized by the number $d$ of item sizes.