A Tight Double-Exponentially Lower Bound for High-Multiplicity Bin Packing

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 all information from a 3-SAT instance with $n$ variables into an ILP with $O(\log(n))$ variables and constraints. This result confirms that the Goemans and Rothvoss algorithm is essentially best-possible for Bin Packing parameterized by the number $d$ of item sizes in the context of XP time algorithms.