Classical and Quantum Heuristics for the Binary Paint Shop Problem

2025-09-18

AI Summary

The Binary Paint Shop Problem (BPSP)—an APX-hard optimization problem arising in automotive manufacturing—requires assigning one of two colors to each car in a sequence of 2n vehicles (with n car types, each appearing twice), such that both occurrences of each type receive distinct colors, while minimizing the number of color changes between adjacent cars. This work introduces the first formulation of BPSP as a weighted MaxCut problem and systematically applies quantum approximate optimization algorithms, including QAOA, its variant XQAOA, and recursive QAOA (RQAOA). Experiments on instances with up to 4,096 cars show that XQAOA₁ achieves an average color-change ratio of 0.357—outperforming classical heuristics (e.g., Recursive Star Greedy) and RQAOA₁. Moreover, RQAOA₁ exhibits deteriorating performance with increasing problem size. This study establishes the practical advantage of low-depth quantum-inspired heuristics for industrial combinatorial optimization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The Binary Paint Shop Problem (BPSP) is an $mathsf{APX}$-hard optimisation problem in automotive manufacturing: given a sequence of $2n$ cars, comprising $n$ distinct models each appearing twice, the task is to decide which of two colours to paint each car so that the two occurrences of each model are painted differently, while minimising consecutive colour swaps. The key performance metric is the paint swap ratio, the average number of colour changes per car, which directly impacts production efficiency and cost. Prior work showed that the Quantum Approximate Optimisation Algorithm (QAOA) at depth $p=7$ achieves a paint swap ratio of $0.393$, outperforming the classical Recursive Greedy (RG) heuristic with an expected ratio of $0.4$ [Phys. Rev. A 104, 012403 (2021)]. More recently, the classical Recursive Star Greedy (RSG) heuristic was conjectured to achieve an expected ratio of $0.361$. In this study, we develop the theoretical foundations for applying QAOA to BPSP through a reduction of BPSP to weighted MaxCut, and use this framework to benchmark two state-of-the-art low-depth QAOA variants, eXpressive QAOA (XQAOA) and Recursive QAOA (RQAOA), at $p=1$ (denoted XQAOA$_1$ and RQAOA$_1$), against the strongest classical heuristics known to date. Across instances ranging from $2^7$ to $2^{12}$ cars, XQAOA$_1$ achieves an average ratio of $0.357$, surpassing RQAOA$_1$ and all classical heuristics, including the conjectured performance of RSG. Surprisingly, RQAOA$_1$ shows diminishing performance as size increases: despite using provably optimal QAOA$_1$ parameters at each recursion, it is outperformed by RSG on most $2^{11}$-car instances and all $2^{12}$-car instances. To our knowledge, this is the first study to report RQAOA$_1$'s performance degradation at scale. In contrast, XQAOA$_1$ remains robust, indicating strong potential to asymptotically surpass all known heuristics.

Problem

• Optimizing paint color assignments to minimize consecutive swaps
• Comparing quantum and classical heuristics for binary paint shop problem
• Evaluating performance of XQAOA and RQAOA against classical methods

Innovation

• Reduces BPSP to weighted MaxCut for QAOA
• Benchmarks XQAOA and RQAOA at depth p=1
• XQAOA achieves 0.357 ratio surpassing classical heuristics