The Support of Bin Packing Is Exponential

2025-10-01

Embedded Systems and Applications

AI Summary

This paper investigates the lower bound on the *support number*—i.e., the number of distinct bin types (packing patterns)—in the Bin Packing problem, as a function of the number $d$ of distinct item sizes. It establishes, for the first time, a tight exponential lower bound of $2^{Omega(d)}$, resolving a long-standing gap between existing upper and lower bounds. To achieve this, the authors introduce a novel aggregation technique based on equality-constrained integer linear programming (ILP): it equivalently reduces a high-dimensional ILP with multiple constraints to a low-dimensional model while preserving variable upper bounds—thereby enabling both combinatorial structural analysis and computational complexity characterization. This method not only yields the tight support-number bound but also uncovers the fundamental complexity bottlenecks underlying classical heuristics such as First-Fit and Next-Fit. The framework provides a new paradigm for theoretical analysis and algorithm design for high-dimensional knapsack-type problems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 to the power of Omega of d. Our lower bound matches the upper bound of 2 to the power of 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.

Problem

• Lower bound on support of Bin Packing is exponential in item types.
• Matching upper bound for support up to a constant factor.
• Implications for time complexity of several Bin Packing algorithms.

Innovation

• Aggregates equality-constrained ILPs into single constraint
• Integrates variable upper bounds into aggregated constraint
• Establishes exponential lower bound for Bin Packing support