High-Multiplicity Bin Packing Complexity

• High-multiplicity bin packing is a 1-dimensional cutting stock problem that packs multiple copies of distinct items into fixed-capacity bins, with sizes and counts encoded in binary.
• Recent research establishes sharp doubly exponential lower bounds on time complexity in terms of the number of item types, confirming inherent computational challenges under the ETH.
• Polyhedral and ILP formulations, leveraging structure theorems and pattern enumeration, offer crucial insights despite the exponential support requirements that limit efficient solution strategies.

High-multiplicity bin packing, also known as the d-type or 1-dimensional cutting stock problem, concerns the optimal packing of multiple copies of a small set of distinct item types into the fewest possible bins of fixed capacity. Each item type is specified by its integer size and multiplicity, both encoded in binary, and the central computational question is how the number of item types $d$ governs the complexity and structure of the problem. Recent research has established sharp exponential lower bounds on both algorithmic time complexity and solution support, confirming that classical approaches cannot avoid double-exponential dependence on $d$ under the Exponential Time Hypothesis (ETH).

1. Formal Problem Definition

A high-multiplicity bin packing instance $I$ is parameterized by:

• $d > 0$: number of distinct item types
• $\mathbf{s} = (s_1, \ldots, s_d)$, $s_j \in \mathbb{N}$, $1 \leq s_j \leq B$: size vector of item types
• $\mathbf{a} = (a_1, \ldots, a_d)$, $a_j \in \mathbb{N}$: multiplicities
• $B \in \mathbb{N}$: bin capacity

The goal is to find the smallest $k \in \mathbb{N}$ such that there exist non-negative integer vectors $x^{(1)}, \ldots, x^{(k)} \in \mathbb{N}^d$ (representing the packing of each bin) with: $$\sum_{j=1}^d s_j \cdot x_j^{(t)} \leq B\quad \forall t=1,\ldots,k$$ and

$\sum_{t=1}^k x^{(t)} = a$

Defining $\Delta = \max\{B, \max_j s_j, \max_j a_j\}$, the encoding length is $|I| = O(d \cdot \log \Delta)$ (Jansen et al., 2 Dec 2025, Goemans et al., 2013).

2. Algorithmic Landscape and Complexity

Early algorithms for high-multiplicity bin packing focused on exact solutions for fixed $d$. The Goemans–Rothvoss algorithm reduces the problem to Cone ∩ Polytope membership in $\mathbb{R}^d$, yielding runtime: $|I|^{2^{O(d)}}$ which is polynomial for constant $d$, but doubly exponential in $d$ in general (Jansen et al., 2 Dec 2025, Goemans et al., 2013). Jansen–Klein leverage the vertex structure of the integer knapsack polytope to improve the dependence in special cases—if the number of vertices $V_I$ is polynomially bounded—but in the worst case $V_I = \Delta^{2^{O(d)}}$, so the doubly exponential barrier persists.

A comprehensive table of algorithmic results:

Algorithm (Reference)	Time Complexity	Notes
Goemans–Rothvoss (2014/2020)	$|I|^{2^{O(d)}}$	Polytime for fixed $d$
Jansen–Klein (2017)	$|V_I|^{O(d)} \cdot \text{poly}(\log\Delta)$	Special cases; $V_I$ can be large
Jansen–Solis-Oba (2011) (approx)	$d^{O(d2^d)} \cdot 2^{O(8^d)}$	OPT+1 additive approximation

These results illustrate that all known approaches ultimately suffer from an inherent exponential blow-up in $d$ (Jansen et al., 2 Dec 2025, Jansen et al., 2 Dec 2025).

3. Lower Bound Constructions and Complexity Theory

A tight doubly exponential lower bound has been proved for high-multiplicity bin packing under ETH. Specifically, unless ETH fails, there is no algorithm for high-multiplicity bin packing with $d$ item types running in time

$|I|^{2^{o(d)}}$

This establishes the optimality of the Goemans–Rothvoss algorithm in the general case and confirms that no improvement of the form $f(d)\,\text{poly}(\log\Delta)$ is possible without breaking ETH (Jansen et al., 2 Dec 2025).

The proof centers on a reduction from 3-SAT, compactly encoding the assignment structure of a large instance into a high-multiplicity bin-packing instance with $d = O(\log n)$, where $n$ is the number of SAT variables. Core technical steps involve sophisticated ILP encodings and multi-level "base-M aggregation" that preserve solution patterns. The constructed instance enforces that finding a packing with a prescribed number of bins is equivalent to finding a satisfying assignment, and the bin packing instance encoding length is polynomially related to the 3-SAT instance size (Jansen et al., 2 Dec 2025).

4. Polyhedral and ILP Formulations

The problem can be recast in geometrical and integer programming terms. The pattern polytope: $P = \{ x \in \mathbb{R}_+^d\mid s^T x \leq B \}$ and the covering constraint $\sum_{x \in P \cap \mathbb{Z}^d} \lambda_x x = a$, with integer optimization variable $\lambda_x$, encapsulate the feasible assignments.

A foundational result is the “Structure Theorem”: for any fixed $d$, there exists a small precomputable set $X \subseteq P \cap \mathbb{Z}^d$ such that any feasible integer assignment from the integer cone over $P$ can be written as a combination with at most $2^{2d}$ points from $X$, and up to $2^{2d}$ additional points outside $X$ used with multiplicity 0 or 1 (Goemans et al., 2013). Lenstra’s algorithm is then applied to solve the fixed-dimension ILP arising from support-based enumeration.

When the polytope's integer hull has exponential in $d$ vertices, as is generically the case after aggregation, even strong polyhedral techniques do not circumvent the exponential enumeration barrier (Jansen et al., 2 Dec 2025).

5. Exponential Support Bounds

A distinctive aspect of the high-multiplicity regime is the support—the number of distinct bin types appearing in an optimal solution. It has been shown that there exist instances with

$\mathrm{supp}^*(d) \geq 2^{\Omega(d)}$

This matches previous upper bounds and demonstrates that any pattern-based ILP or combinatorial algorithm must confront 2$^d$-scale support (Jansen et al., 2 Dec 2025). The construction involves encoding a knapsack polytope whose integer points cannot be partitioned or combined except with one occurrence of each, enforced via tailored aggregation of multiple equality constraints into a single knapsack constraint with preserved solution space. This directly precludes polynomial or even single-exponential support bounds.

Practical implication: listing all necessary patterns for pattern-based ILP or dynamic programming methods incurs exponential time and space.

6. Generalizations and Related Problems

The polyhedral approach and complexity barriers extend directly to related high-multiplicity resource allocation frameworks, such as the cutting stock problem, the scheduling of jobs on machines with a fixed number of job or machine types, and multidimensional generalizations. For each, the exponential support lower bound and the doubly exponential time complexity dependence on $d$ (number of item types or resource types) carry over (Goemans et al., 2013, Jansen et al., 2 Dec 2025).

Corollaries include:

• For multiple bin types ($m$), the complexity is $(\log\Delta)^{2^{O(d+m)}}$.
• For high-multiplicity scheduling on a fixed number of machine types, polynomial-time solutions exist only if both $d$ and $m$ are fixed.

7. Implications and Barriers for Parameterized and Exact Algorithms

The combinatorial and algorithmic barriers established preclude fixed-parameter tractability (FPT) of the form $f(d) \,\text{poly}(|I|)$ with $f(d)$ subexponential in $d$, unless ETH fails. These results further show that improvements must depart fundamentally from patterns, cone enumeration, or small-support strategies—any approach relying on ILP or configuration enumeration is inherently limited. The open problem remains for special cases or restricted input classes, but the generic case is now tightly classified (Jansen et al., 2 Dec 2025, Jansen et al., 2 Dec 2025).

A plausible implication is that advances will require fundamentally different paradigms, rather than refinements of existing polyhedral or combinatorial methods.

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References

• "A Tight Double-Exponentially Lower Bound for High-Multiplicity Bin Packing" (Jansen et al., 2 Dec 2025)
• "Polynomiality for Bin Packing with a Constant Number of Item Types" (Goemans et al., 2013)
• "The Support of Bin Packing is Exponential" (Jansen et al., 2 Dec 2025)