Goemans & Rothvoss Bin Packing Algorithm

• The paper presents a novel reduction of high-multiplicity bin packing to finding minimal integer conic combinations in a d-dimensional polytope.
• It leverages strategic enumeration of feasible configurations and a key combinatorial lemma limiting solutions to at most 2^d nonzero patterns.
• The algorithm attains optimal running time bounds under ETH, confirming that no fixed-parameter tractable solution exists solely by parameter d.

The Goemans and Rothvoss Algorithm refers to a pivotal algorithmic framework and complexity result for the high-multiplicity bin packing problem, formally introduced by Goemans and Rothvoss in 2014. It establishes the best-possible running time (as a function of the number of distinct item types) for solving bin packing when items are supplied in quantities given in binary encoding. This result resolves a long-standing question concerning the tractability of high-multiplicity bin packing with respect to the parameter $d$, the number of item types.

1. Problem Definition and Formalization

Consider $d$ types of items, with each type $i$ having a size $s_i \in (0, C]$ and multiplicity $a_i \in \mathbb{Z}_{\ge 0}$. The input specifies these sizes and multiplicities, typically in binary encoding, and a bin with capacity $C$. A valid packing corresponds to a multiset of vectors $x^{(1)},\ldots,x^{(B)} \in \mathbb{Z}_{\ge 0}^d$, each satisfying $s^T x^{(j)} \leq C$. The total assignment must satisfy the demand constraint $\sum_{j=1}^B x^{(j)} = a$.

A solution can alternatively be captured as a non-negative integer vector $x \in \mathbb{Z}_{\ge 0}^{|\mathcal{C}|}$ over the set $\mathcal{C}$ of feasible patterns (configurations), where the support $\text{supp}(x)$ is the number of distinct patterns used. The high-multiplicity bin packing asks to minimize the total bins $B$ needed, given $(s, a, C)$.

2. Algorithmic Principles and Main Result

Goemans and Rothvoss reduce bin packing for fixed $d$ to the problem of finding a minimal integer conic combination in a $d$-dimensional polytope. Specifically, defining

$$\mathcal{P} = \{x \in \mathbb{Z}_{\ge 0}^d : s^T x \leq C\}$$

the goal is to find non-negative integers $\{\lambda_x\}_{x \in \mathcal{P}}$ so that

$\sum_{x \in \mathcal{P}} \lambda_x x = a,$

minimizing $\sum_{x} \lambda_x$.

A key combinatorial lemma of Eisenbrand and Shmonin (2006) establishes any $y \in \mathbb{Z}^d$ in the integer cone of $P \cap \mathbb{Z}^d$ can be expressed with at most $2^d$ nonzero configurations. Consequently, Goemans and Rothvoss obtain an algorithm that, for fixed $d$, runs in time

$(\log \Delta)^{2^{O(d)}}$

where $\Delta$ denotes data encoding size. This result confirms the problem is polynomial-time solvable in the length of input for fixed $d$ and matches the best known structural bound on support size (Goemans et al., 2013).

3. Methodological Innovations

The methodology proceeds by enumerating all configurations $X$ (the "core") of size polynomial in input for fixed $d$, such that any optimal solution uses at most $2^d$ patterns outside $X$, each at most once. For each candidate support, an integer program with at most $2^d$ variables is solved using classic Lenstra-type approaches.

The crucial structural fact is that for any valid packing, only a doubly exponential number of configurations (in $d$) need to be considered. This leads to a compressed search space for the solution.

A notable aggregation technique transforms multiple equality-constrained ILPs into a single knapsack constraint using a "base-$M$" trick, allowing the entire problem to be encoded as a single-constraint ILP, suitable for reduction to bin packing with derived item sizes, multiplicities, and bin capacity (Jansen et al., 2 Dec 2025).

4. Complexity Lower Bounds and Optimality

Jansen, Pirotton, and Tutas (Jansen et al., 2 Dec 2025) demonstrate a matching lower bound: the minimum support of optimal solutions can be $2^{\Omega(d)}$, showing that the Eisenbrand–Shmonin bound is tight up to constants. Furthermore, Jansen, Ohnesorge, and Pirotton (Jansen et al., 2 Dec 2025) prove (assuming the Exponential Time Hypothesis) that no algorithm can solve high-multiplicity bin packing in time $|I|^{2^{o(d)}}$, where $|I|$ is the encoding length. Thus, the doubly exponential dependence on $d$ in the exponent is necessary.

This validates the essential optimality of the Goemans and Rothvoss algorithm with respect to $d$, affirmatively resolving whether polynomial or singly-exponential (in $d$) time FPT algorithms existed for this problem—they do not, barring unexpected complexity-theoretic collapses.

5. Comparative Analysis and Related Algorithms

Subsequent works (notably Jansen–Klein [SODA ’17]) provide improvements in runtime under further structural assumptions, particularly when the integer knapsack polytope has few vertices. In the special case where $|V_I|$ (the number of polytope vertices) is small, they achieve runtimes of $|V_I|^{2^{O(d)}} \cdot \mathrm{poly}(\mathrm{enc})$. However, $|V_I|$ can be as large as $\Delta^{2^{O(d)}}$, so this does not break the double-exponential-in-$d$ barrier in the worst case (Jansen et al., 2 Dec 2025).

The table below summarizes key algorithms, their running times, and structural bounds:

Algorithm	Running Time	Support Bound Used
Goemans–Rothvoss (2014)	$|I|^{2^{O(d)}}$	$2^d$
Jansen–Klein (2017, special cases)	$|V_I|^{2^{O(d)}} \mathrm{poly}(\mathrm{enc})$	$2^d$
Jansen–Solis-Oba (2010)	$d^{O(d2^d)} 2^{O(8^d)}\mathrm{poly}(\mathrm{enc})$	$2^d$

6. Implications and Future Directions

The Goemans and Rothvoss algorithm represents a canonical result in parameterized and exact algorithms for combinatorial optimization with high-multiplicity. The matching lower bounds preclude expectation of an FPT (fixed-parameter tractable) exact algorithm parameterized solely by $d$ unless the ETH fails.

Consequently, advances are only possible by introducing additional parameters (e.g., bounding $|V_I|$), exploiting further instance structure, or moving to approximation settings. The aggregation trick for encoding multiple constraints as a single constraint also impacts the paper of multi-dimensional knapsack and general integer linear programming.

A plausible implication is that any significant future breakthrough on this front will derive from improved understanding of instance-specific structure, approximation, or hybrid methods sensitive to the geometric structure of configuration spaces.

7. Historical and Research Context

M. X. Goemans and T. Rothvoss formalized the approach and computational complexity in SODA ’14 and J. ACM ’20, building essentially on the support bounds of Eisenbrand and Shmonin. Recent works (Jansen et al., 2 Dec 2025, Jansen et al., 2 Dec 2025) fully resolve the central open questions on support-minimality and running time lower bounds, closing an important gap in the literature. High-multiplicity bin packing, cutting stock, and high-multiplicity scheduling remain active topics, particularly with the focus on extended formulations, ILP aggregation, and encoding barriers in resource allocation problems.