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

TL;DR

This paper establishes a tight lower bound showing that high-multiplicity Bin Packing cannot be solved faster than a double-exponential in the number of item types unless ETH fails, confirming the optimality of existing algorithms.

Contribution

It proves that the known algorithm's doubly exponential runtime dependency on item types is essentially optimal under ETH, resolving a major open problem.

Findings

No faster algorithm than the existing double-exponential bound unless ETH fails.

Introduces a novel reduction from 3-SAT encoding into ILP with logarithmic variables.

Confirms the optimality of the Goemans and Rothvoss algorithm for high-multiplicity Bin Packing.

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 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.