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Classical and Quantum Heuristics for the Binary Paint Shop Problem

V Vijendran, Dax Enshan Koh, Ping Koy Lam, Syed M Assad

TL;DR

This work develops a rigorous ICC-based Ising reduction of the Binary Paint Shop Problem (BPSP) to a weighted MaxCut formulation, enabling efficient quantum-heuristic evaluation on near-term devices. By benchmarking two low-depth QAOA variants, XQAOA$_1$ and RQAOA$_1$, against strong classical heuristics, the study demonstrates that XQAOA$_1$ achieves a robust average paint-swap ratio of roughly $0.357$ across instance sizes up to $2^{12}$ cars, outperforming all classical methods and remaining stable as problem size grows. In contrast, RQAOA$_1$ exhibits size-dependent deterioration, ultimately being surpassed by classical heuristics, highlighting that expressiveness in the QAOA ansatz can trump non-locality at scale. The results suggest that XQAOA$_1$ holds significant potential to asymptotically surpass known heuristics for BPSP, and they motivate further exploration of expressively rich, shallow quantum circuits for industrial combinatorial tasks, including extensions to higher-depth, multi-colour variants, and related scheduling problems. The paper also strengthens the methodological bridge between BPSP and MaxCut via the ICC encoding, offering a blueprint for applying QAOA-like heuristics to other Ising-encodable problems.

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.

Classical and Quantum Heuristics for the Binary Paint Shop Problem

TL;DR

This work develops a rigorous ICC-based Ising reduction of the Binary Paint Shop Problem (BPSP) to a weighted MaxCut formulation, enabling efficient quantum-heuristic evaluation on near-term devices. By benchmarking two low-depth QAOA variants, XQAOA and RQAOA, against strong classical heuristics, the study demonstrates that XQAOA achieves a robust average paint-swap ratio of roughly across instance sizes up to cars, outperforming all classical methods and remaining stable as problem size grows. In contrast, RQAOA exhibits size-dependent deterioration, ultimately being surpassed by classical heuristics, highlighting that expressiveness in the QAOA ansatz can trump non-locality at scale. The results suggest that XQAOA holds significant potential to asymptotically surpass known heuristics for BPSP, and they motivate further exploration of expressively rich, shallow quantum circuits for industrial combinatorial tasks, including extensions to higher-depth, multi-colour variants, and related scheduling problems. The paper also strengthens the methodological bridge between BPSP and MaxCut via the ICC encoding, offering a blueprint for applying QAOA-like heuristics to other Ising-encodable problems.

Abstract

The Binary Paint Shop Problem (BPSP) is an -hard optimisation problem in automotive manufacturing: given a sequence of cars, comprising 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 achieves a paint swap ratio of , outperforming the classical Recursive Greedy (RG) heuristic with an expected ratio of [Phys. Rev. A 104, 012403 (2021)]. More recently, the classical Recursive Star Greedy (RSG) heuristic was conjectured to achieve an expected ratio of . 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 (denoted XQAOA and RQAOA), against the strongest classical heuristics known to date. Across instances ranging from to cars, XQAOA achieves an average ratio of , surpassing RQAOA and all classical heuristics, including the conjectured performance of RSG. Surprisingly, RQAOA shows diminishing performance as size increases: despite using provably optimal QAOA parameters at each recursion, it is outperformed by RSG on most -car instances and all -car instances. To our knowledge, this is the first study to report RQAOA's performance degradation at scale. In contrast, XQAOA remains robust, indicating strong potential to asymptotically surpass all known heuristics.

Paper Structure

This paper contains 36 sections, 16 theorems, 96 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Binary Paint Shop Problem
  4. Quantum Approximate Optimisation Algorithm
  5. Expressive QAOA
  6. Recursive QAOA
  7. Recasting BPSP as an Ising Model
  8. Initial Car Colour Encoding
  9. BPSP Graph Construction
  10. Reduction to Weighted MaxCut
  11. Ising Model Formulation
  12. Numerical Results
  13. Discussions
  14. Diminishing Returns of Level-1 RQAOA
  15. Performance of Level-1 XQAOA as n to infinity
  16. ...and 21 more sections

Key Result

Lemma 2

Let $x \in \Gamma_n$ and $i \in [2n-1]$. If $x_i = x_{i+1}$, then $\eta(x, i) = 1$.

Figures (3)

  • Figure 1: Solving the BPSP on a Quantum Computer Using the ICC Encoding. This figure illustrates the process of solving the BPSP with the Initial-Car-Colour (ICC) encoding using QAOA and its variants. a) The process starts with a BPSP instance, which in this example consists of 6 cars of length 12 forming the sequence C$_5$C$_1$C$_1$C$_3$C$_2$C$_2$C$_5$C$_4$C$_3$C$_6$C$_6$C$_4$. b) Using the ICC encoding the BPSP instance with 6 cars is mapped to an Ising model with 6 spins. In the depicted graphs, orange edges symbolise an edge weight of +1, while green edges indicate an edge weight of -1. c) The Ising model is then solved on quantum computer using QAOA or its variants. d) This is the ground state solution of the Ising model, where the colour of node (based on the spin) determines the colour of the first occurrence of the car; using this colour the second occurrence of the car is painted the opposite colour. e) This is the solved BPSP instance with a total of 4 paint swaps.
  • Figure 2: Benchmarking Classical and Quantum Heuristics on BPSP Instances. (a) Line plot showing the number of paint swaps produced by different heuristics on 200 random problems for each $n \in \{5,10,\ldots,120\}$. Each point represents the average over all instances, with XQAOA$_1$ evaluated from the best of 100 random restarts per problem. Among the classical heuristics, Red-First performs the worst, while RSG performs the best. Both quantum heuristics achieve lower swap counts, with XQAOA$_1$ consistently outperforming RQAOA$_1$. (b) Boxplot showing the distribution of paint swap ratios for larger instances comprising 50 random problems for each $n \in \{128,256,512,1024,2048,4096\}$. For XQAOA$_1$, the boxplots include all solutions from 100 restarts per instance. RSG achieves the best classical performance with an average ratio of about $0.37$, yet it is consistently surpassed by XQAOA$_1$, which attains an average ratio of roughly $0.357$. RQAOA$_1$ performs well for smaller $n$, but its advantage diminishes after $n=512$. The best solutions from XQAOA$_1$, visible as outliers below the whiskers, consistently outperform those of RQAOA$_1$. (c–e) Scatterplots comparing the relative performance of RSG, RQAOA$_1$, and XQAOA$_1$. Each plot places one algorithm on each axis, with the $y=x$ diagonal as reference: points above the diagonal indicate superior performance of the $x$-axis algorithm, while points below indicate the $y$-axis algorithm performs better. Panel (c) shows RQAOA$_1$ outperforming RSG on most instances up to $n=1024$, after which RSG dominates, particularly for $n \geq 2048$. Panel (d) shows XQAOA$_1$ consistently outperforming RSG by a wide margin across all sizes. Panel (e) shows XQAOA$_1$ outperforming RQAOA$_1$ in nearly all cases, with only a few exceptions.
  • Figure 3: Distribution of Paint Swap Ratios for RQAOA$_1$ and XQAOA$_1$. This figure shows KDE plots of paint-swap ratio distributions for larger instances comprising 50 random problems for each $n \in \{128,256,512,1024,2048,4096\}$. Panel (a) presents RQAOA$_1$ and panel (b) XQAOA$_1$, with the latter including all solutions from 100 restarts per instance. For RQAOA$_1$, the mean of each distribution shifts steadily right as $n$ increases, indicating deteriorating performance: rising from 0.322 at $n=128$ to 0.362 at $n=1024$, where it already falls behind XQAOA$_1$’s average of 0.357. At $n=2048$, the mean increases further to 0.375, worse than RSG’s 0.37, and by $n=4096$ it reaches 0.39, with the right tail extending to 0.4, matching the performance of RG. In contrast, XQAOA$_1$ exhibits stable behaviour across all sizes, with means consistently near 0.357. As $n$ grows, its distributions sharpen and concentrate around this value, strongly suggesting that XQAOA$_1$ outputs solutions with an average swap ratio of about 0.357 in the large limit.

Theorems & Definitions (32)

  • Conjecture 1
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 22 more