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

2025-12-02

AI Summary

This paper investigates the computational complexity of high-multiplicity bin packing, focusing on the dependence of optimal algorithm running time on the number of item types $d$. We establish a tight reduction from 3-SAT to an integer linear program (ILP) with only $O(log n)$ variables, achieving the first efficient encoding of $n$-variable logical information into a logarithmic-size ILP. This yields a lower bound: unless the Exponential Time Hypothesis (ETH) fails, no algorithm can solve the problem in time $|I|^{2^{o(d)}}$. The bound is tight, confirming that the double-exponential-time algorithm by Goemans and Rothvoß (2014) is optimal—thereby resolving their open question. Our approach integrates techniques from parameterized complexity, fine-grained reductions under ETH, and combinatorial circuit encoding.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Proves double-exponential lower bound for high-multiplicity bin packing runtime
• Shows no algorithm can avoid exponential dependency on distinct item types
• Confirms optimality of existing algorithm for bin packing parameterized by item sizes

Innovation

• Novel reduction from 3-SAT to Bin Packing
• Efficient encoding of 3-SAT into ILP with O(log n) variables
• Proves double-exponential dependency on d is necessary