Formulating Multistage Cutting Stock Problems as QUBO
Marcel Seelbach Benkner, Chiara Capecci, Sebastian Nagies, Javed Akram, Sebastian Rubbert, Dimitrios Bantounas, Philipp Hauke, Michael Johanning, Michael Moeller
quant-phmath.OCSubmitted Sep 12, 2026• Announced Sep 15, 2026• 0 citations
Cutting stock problems are of large relevance to a variety of industry branches. Here, we present a linear programming formulation for a restricted version of the multistage 2D cutting stock problem with Guillotine cuts and test it on published benchmark instances. After this, we derive a non-exact reformulation in quadratic unconstrained binary optimization (QUBO) form via unbalanced penalization, which we show to have possible benefits in modeling the problem with usable leftovers. This work mainly considers the restricted formulation, in which the dimensions of each cut are determined by those of a single required piece. We further discuss how the approach can be extended to the unrestricted problem, which allows more general cutting patterns involving cuts with dimensions of multiple required pieces and naturally gives rise to quadratic terms in the inequalities of the problem formulation. We showcase solutions via a Simulated Annealing solver for the restricted problem class, where we mainly study the performance of the iterative augmented Lagrangian method. We also discuss possible applications of Quantum Annealing in this setting, as a strong motivation to research QUBO formulations.
Cutting stock problems are of large relevance to a variety of industry branches. Here, we present a linear programming formulation for a restricted version of the multistage 2D cutting stock problem with Guillotine cuts and test it on published benchmark instances. After this, we derive a non-exact reformulation in quadratic unconstrained binary optimization (QUBO) form via unbalanced penalization, which we show to have possible benefits in modeling the problem with usable leftovers. This work mainly considers the restricted formulation, in which the dimensions of each cut are determined by those of a single required piece. We further discuss how the approach can be extended to the unrestricted problem, which allows more general cutting patterns involving cuts with dimensions of multiple required pieces and naturally gives rise to quadratic terms in the inequalities of the problem formulation. We showcase solutions via a Simulated Annealing solver for the restricted problem class, where we mainly study the performance of the iterative augmented Lagrangian method. We also discuss possible applications of Quantum Annealing in this setting, as a strong motivation to research QUBO formulations.
The two-dimensional Cutting Stock Problem (2D-CSP) is an NP-hard problem with direct economic and environmental impacts on manufacturing and logistics. We encode its fixed-plate variant, with free piece repetition and full non-overlap and containment constraints, as a Quadratic Unconstrained Binary Optimization (QUBO) problem for quantum annealing and compare two formulations from opposite encoding paradigms. The first was a coordinate-based set-packing model with one binary variable per candidate placement. Its variable count grows linearly with plate area and resolution, but its ground state is, by construction, a geometrically feasible maximum-area packing. The second is a coordinate-free sequence-pair model whose variable count is independent of plate resolution and size. We prove that this compactness has a structural limit: no coordinate-free QUBO of bounded interaction degree whose penalties vanish on every geometrically feasible layout can have a geometrically feasible ground state for 2D containment, because containment is a longest-path constraint that bounded-degree penalties cannot enforce on chains longer than their interaction order. We evaluate both formulations under multi-seed simulated annealing, simulated quantum annealing, and an exact integer-programming baseline. Hardware experiments include D-Wave minor embedding, a calibrated direct-QPU sweep, and Leap hybrid solvers in both penalty and constraint-native form, across a six-instance campaign with per-instance calibration. The hybrid solver returns our certificate configuration, tying its energy to thirteen decimal places while overflowing the plate. We claim no quantum speedup. Our contribution is an impossibility result characterizing the limits of compact packing QUBOs, and a practical rule for choosing between the two formulations.
Abstract Quadratic unconstrained binary optimization (QUBO) formulations of combinatorial optimization problems allow for solving them using various quantum heuristics. While large-scale quantum computations are currently still out of reach, we can already numerically test such QUBO formulations on a perhaps surprisingly large scale. In this work, we benchmark Fujitsu’s Digital Annealer (DA) on the Max-Cut problem, which captures the main complexity of the QUBO problem. We make a comprehensive benchmark against leading other heuristic algorithms on graphs with up to 53,000 variables by focusing on the wall-clock time. Moreover, we compare the DA performance against published performance results of the D-Wave hybrid quantum–classical annealer and the recently proposed QIS3 heuristic. Based on performance statistics for over 2000 graphs from the MQLib, we find that the DA yields competitive results. We hope that this benchmark demonstrates the extent to which large QUBO instances can be heuristically solved today, yielding consistent results across different solvers.
Quadratic unconstrained binary optimization (QUBO) provides problem formulations for various computational problems that can be solved with dedicated QUBO solvers, based on classical or quantum computation. A common approach to constrained combinatorial optimization problems is to enforce the constraints in the QUBO formulation by adding penalization terms. Penalization introduces an additional hyperparameter that significantly affects the solver's efficacy: the relative weight between the objective terms and the penalization terms. Here we study the problem of a priori determining suitable penalization weights from a foundational and practical perspective. We formally establish that finding its optimal value for Gibbs solvers is NP-hard even approximately and #P-hard exactly. This serves us as a motivation to develop a widely applicable heuristic strategy for determining suitable penalization weights, with provable guarantees for Gibbs solvers, and polynomial complexity for broad problem classes. Experiments across diverse problems and solver architectures, including large-scale instances on Fujitsu's Digital Annealer, show robust performance and order-of-magnitude speedups over existing heuristics. We further extend the strategy to quantum approximate optimization and demonstrate its performance on Quantinuum's H2-2 trapped-ion processor.
The development of quantum annealing has stimulated interest in solving NP-hard problems, including various industrial problems, such as quadratic unconstrained binary optimization (QUBO), with specialized solvers. This study focuses on the rectangle packing problem (RPP), which is an NP-hard problem applicable to industrial applications. The objective of this problem is to place given items in a minimum area without overlap. Naive QUBO representations, such as a direct binary encoding of locations, could require a large number of bits, which limits the problem size, hindering the acquisition of the optimal solution. To circumvent this difficulty, we employ the concept of sequence pairs to efficiently represent the locations of items. We also propose a strategy that uses multiple sampling solutions obtained with a QUBO solver, where we employ new constraints on the number of the rectangles contributing to the placement area for restricting the search space to smoothly bridge the QUBO formulation of the sequence pair and the minimization of area of rectangle placement. We estimate an approximate curve for the optimum number of rectangles contributing to the placement area. Then, we add the penalty term concerning the deviation from the estimated optimal value on the objective function. In numerical experiments with instances with 4 to 10 rectangles, we demonstrate that the proposed sampling-based strategy can efficiently find feasible solutions. This strategy would be useful for dealing with more complex problems, such as rectangle orientation and three-dimensional problems, with a relatively small number of bits using sequence pairs.
Quadratic Unconstrained Binary Optimization (QUBO) provides a versatile framework for representing NP-hard combinatorial problems, yet existing solvers often face trade-offs among speed, accuracy, and scalability. In this work, we introduce a quantum-inspired solver (QIS) that unites branch-and-bound pruning, continuous gradient-descent refinement, and quantum-inspired heuristics within a fully adaptive control architecture. We benchmark QIS3 against eight state-of-the-art solvers, including genetic algorithms, coherent Ising machines, simulated bifurcation, parallel tempering, simulated annealing, our prior QIS2 version, D-Wave's simulated-annealing (Neal), and Gurobi on three canonical QUBO problem classes: Max-Cut, NAE-3SAT, and Sherrington-Kirkpatrick spin glass problems. Under a uniform runtime budget, QIS3 attains the best solution on nearly all instances, achieving optimality in 94% of max-cut instances. These results establish QIS3 as a robust, high-performance solver that bridges classical exact strategies and quantum-inspired heuristics for scalable QUBO optimization.
This paper addresses the Pickup and Delivery Problem with Time Windows (PDPTW), an NP-hard combinatorial optimization problem with major practical relevance in logistics and transportation. The study focuses on a quadratic unconstrained binary optimization (QUBO) formulation for quantum annealing and benchmarks it against two classical optimization paradigms. A modular Python framework is developed that encodes PDPTW in three ways: a mixed-integer linear programming (MILP) model that serves as an exact reference, a genetic algorithm (GA) metaheuristic, and a QUBO model that is compatible with quantum annealers. The framework supports test scenarios with increasing structural complexity, with both feasible and intentionally infeasible instances. An additional contribution is the conceptual design and preliminary analysis of an automatic-penalty weight-tuning scheme for the QUBO model. Experimental results show that the proposed QUBO formulation can produce high-quality solutions for simpler PDPTW instances, but its performance strongly depends on the careful calibration of penalty weights. MILP provides optimal baselines on small instances but becomes intractable as problem size grows. The GA scales to the largest scenario and finds feasible solutions of reasonable quality, but they are not necessarily optimal. The evaluation also includes a large number of problem instances and runs on IBM Quantum hardware using the Quantum Approximate Optimization Algorithm (QAOA).