QAL-BP: An Augmented Lagrangian Quantum Approach for Bin Packing

TL;DR

The Bin Packing Problem is NP-hard, and this work targets quantum-accelerated solutions by formulating it as a QUBO using an Augmented Lagrangian framework. The proposed QAL-BP end-to-end approach embeds capacity and assignment constraints into the objective with analytically estimated penalty terms, avoiding instance-specific empirical tuning and reducing the number of qubits compared to pseudo-polynomial formulations. Key contributions include a constraint-embedding QUBO that scales more favorably with bin capacity and demonstrated feasibility on a real quantum annealer, with comparisons to simulated annealing and Gurobi showing near-feasible and often optimal performance on tested instances. While current hardware limits prevent QA from outperforming the best classical solver on larger problems, the results suggest promising scaling and potential advantages as quantum devices improve, paving the way for hybrid quantum-classical strategies for larger BPP instances.

Abstract

The bin packing is a well-known NP-Hard problem in the domain of artificial intelligence, posing significant challenges in finding efficient solutions. Conversely, recent advancements in quantum technologies have shown promising potential for achieving substantial computational speedup, particularly in certain problem classes, such as combinatorial optimization. In this study, we introduce QAL-BP, a novel Quadratic Unconstrained Binary Optimization (QUBO) formulation designed specifically for bin packing and suitable for quantum computation. QAL-BP utilizes the Augmented Lagrangian method to incorporate the bin packing constraints into the objective function while also facilitating an analytical estimation of heuristic, but empirically robust, penalty multipliers. This approach leads to a more versatile and generalizable model that eliminates the need for empirically calculating instance-dependent Lagrangian coefficients, a requirement commonly encountered in alternative QUBO formulations for similar problems. To assess the effectiveness of our proposed approach, we conduct experiments on a set of bin packing instances using a real Quantum Annealing device. Additionally, we compare the results with those obtained from two different classical solvers, namely simulated annealing and Gurobi. The experimental findings not only confirm the correctness of the proposed formulation but also demonstrate the potential of quantum computation in effectively solving the bin packing problem, particularly as more reliable quantum technology becomes available.

Figures (8)

• Figure 1: Comparative analysis of variable growth in the Pseudo-Polynomial and Augmented Lagrangian models concerning the number of items and bin capacity. Three distinct values of bin capacity (C) are explored. The continuous dark red line represents the upper limit for QUBO problems represented by fully connected graphs that can be mapped in the D-Wave Advantage Quantum Processing Unit (QPU) equipped with 5640 qubits.
• Figure 2: Comparison of the number of logical variables of QAL-BP against the physical qubits needed to implement the QUBO problem on the D-Wave Advantage 4.1.
• Figure 3: Comparative evaluation of the energy of the solutions provided by SA and QA (top), and the chain_break_fraction reported by the QA implementation (bottom).
• Figure 4: Comparative evaluation of the energy values for different chain break mitigation strategies.
• Figure 5: TTS comparison between Quantum Annealing (QA), Simulated Annealing (SA) and Gurobi. The Gurobi curve ranges between 440$\mu s$ and 3000$\mu s$.