Binary Paint Shop Problem (BPSP)

• BPSP is a combinatorial optimization problem in automotive painting that sequences paired car models to minimize costly consecutive color changes.
• Classical heuristics (RF, RG, RSG) and quantum metaheuristics (QAOA, XQAOA, RQAOA) offer varied strategies to approximate the optimal paint swap ratio.
• Empirical benchmarks and MaxCut reductions underscore BPSP’s industrial relevance and fuel research into quantum and hybrid optimization methods.

The Binary Paint Shop Problem (BPSP) is an APX-hard combinatorial optimization problem originating from automotive manufacturing. It involves sequencing 2n cars, with n distinct models each appearing exactly twice, and assigning one of two permissible paint colors to each appearance so that the two occurrences of any given model are painted differently, while minimizing consecutive color changes (swaps) in the fixed sequence. The minimum attainable number of color changes, or the paint swap ratio (number of color changes per car), is directly tied to production efficiency and cost in industrial paint shops. BPSP's APX-hardness establishes that no classical polynomial-time algorithm can even approximate the optimum within any constant factor, under standard complexity assumptions.

1. Formal Problem Statement and Industrial Context

Given a fixed word $x$ representing a sequence of $2n$ cars (each model appears exactly twice), assign to each position a color $c \in \{\mathrm{red}, \mathrm{blue}\}$ such that for each model, the two occurrences receive different colors. The objective is: $$\text{Minimize} \quad \Delta_C = \sum_{i=1}^{2n-1} \mathbb{I}[c_i \neq c_{i+1}]$$ where $\Delta_C$ gives the total number of color swaps. The core industrial motivation lies in reducing the frequency of paint purges, which are costly both in time and materials. In automotive paint shops, any consecutive color change entails non-negligible overhead, making even incremental improvements in $\Delta_C$ significant at scale.

BPSP is not only NP-complete (decision variant) but APX-hard as an optimization problem—no polynomial-time constant-factor approximation exists assuming the Unique Games Conjecture. These complexity barriers have led industry and academia to focus on heuristic, metaheuristic, and—more recently—quantum and hybrid algorithms for approximate solutions (Streif et al., 2020, Vijendran et al., 18 Sep 2025).

2. Classical Heuristics and Performance Benchmarks

Several classical heuristics have been developed to generate feasible colorings for BPSP:

• Red-First (RF): Assign "red" to each model's first occurrence, "blue" to the second.
• Greedy: At each step, choose the color for the current car occurrence to minimize the immediate increment in color changes, subject to the global constraint.
• Recursive Greedy (RG): Iteratively removes the last-occurring car, solves the reduced instance, and re-inserts the car optimally; expected swap ratio $\approx 0.4$.
• Recursive Star Greedy (RSG): An advanced variant introducing a "star" (undecided) color, resolving ambiguities recursively. RSG is conjectured to reach an expected swap ratio near $0.361$, though empirical averages are closer to $0.37$.

Empirical benchmarks show that red-first coloring performs worst, while RG and RSG heuristics yield the best-known results among classical methods, asymptotically approaching but not achieving the lower bound for the average swap ratio (Vijendran et al., 18 Sep 2025).

3. Quantum Algorithms: QAOA and Variants

The Quantum Approximate Optimization Algorithm (QAOA) provides a quantum metaheuristic framework for BPSP by mapping the problem to an Ising spin model. The reduction involves:

• ICC (Initial Car Colour) Encoding: Compresses the problem from $2n$ variables to $n$ by specifying the color of each model's first occurrence; the sequel is fixed by opposition.
• Ising Hamiltonian Construction:

$$H_P(x) = n - \frac{1}{2} - \frac{1}{2} \sum_{i=1}^{2n-1} (-1)^{\eta(x, i)} Z_{x_i} Z_{x_{i+1}}$$

where $Z_j$ is the Pauli-$Z$ operator for model $j$ and $\eta(x, i)$ encodes local parity (i.e., first/second occurrence).

QAOA seeks to minimize $$\langle \Psi(\{\beta_l, \gamma_l\}) | H_P | \Psi(\{\beta_l, \gamma_l\}) \rangle$$ through a variational quantum circuit of depth $p$, alternating "problem" and "mixing" unitaries.

Two notable low-depth QAOA variants for BPSP:

• eXpressive QAOA (XQAOA): Overparameterized ansatz assigning independent variational angles to each Hamiltonian term and to mixing operators, providing high expressivity at minimal depth.
• Recursive QAOA (RQAOA): Recursively eliminates variables based on pairwise $ZZ$ correlators, constructing solutions via iterative rounding.

Comparison with classical heuristics indicates that quantum algorithms achieve lower or comparable paint swap ratios at moderate depth. For example, QAOA with $p=7$ achieves $0.393$ [Phys. Rev. A 104, 012403 (2021)], outperforming RG; XQAOA$_1$ achieves $0.357$ and is robust to problem scaling, while RQAOA$_1$ degrades on larger instances and can eventually be outperformed by RSG (Vijendran et al., 18 Sep 2025).

4. Problem Reductions, Encoding, and Theoretical Foundations

BPSP can be formally reduced to a weighted MaxCut problem, enabling the use of Ising Hamiltonian-based solution techniques:

• Reduction to MaxCut:
  • The BPSP graph $G_x$ has $n$ vertices (models), with weighted edges derived from adjacent positions in the original sequence ($+1$, $-1$, or $-2$ weights).
  • The objective becomes maximizing the cut-weight induced by the ICC vector $z$: $$\widetilde{\xi}_n(x,z) = \xi_{RF}(x) - \mathrm{wt}_{G_x}(z)$$.

The ICC encoding strictly halves the decision variables: $n$ bits suffice to specify every valid coloring. The Ising mapping allows the application of quantum circuits, where optimization over spin configurations yields optimal or near-optimal color assignments.

5. Empirical Analysis: Performance and Resource Considerations

Detailed studies benchmark quantum and classical methods:

• Paint Swap Ratio Table (asymptotic averages):

Algorithm	Swap Ratio (empirical)	Scaling Observation
Red-First	$\sim$0.5	Linear in $n$
RG	0.4	Matches QAOA $p=3$ and below
RSG	0.37	Conjectured as low as 0.361
QAOA $p=5$	0.432	Outperforms RG
XQAOA$_1$	0.357	Best known for $2^7$ to $2^{12}$
RQAOA$_1$	0.390	Degrades for large $n$

Quantum algorithms, especially XQAOA$_1$, are empirically found to outperform all classical heuristics including RSG for large $n$, with the performance gap widening as problem size increases. However, RQAOA$_1$ underperforms relative to XQAOA$_1$ and RSG for large-scale instances, despite parameter optimization at every recursion (Vijendran et al., 18 Sep 2025).

Resource use in quantum methods, such as circuit depth, CNOT count, entanglement entropy, and bond dimensions, has been analyzed in terms of classical simulability and NISQ device implementability (Mooney et al., 15 Jul 2025). The use of precomputed parameters and reverse-causal-cone measurements in RQAOA leads to dramatically reduced circuit complexity compared with traditional QAOA.

Experiments on trapped-ion quantum hardware have demonstrated near-ideal performance for instances with up to $n=2$, but noise and depth limitations degrade results for $n \geq 11$, approaching the performance of random guessing (Streif et al., 2020).

6. Generalizations and Extensions

The BPSP has been generalized to multi-car paint shop problems (MCPS) and to variants that couple quality and sequence-dependent costs:

• MCPS: Incorporates multiple cars, customer-imposed color quotas, and additional scheduling constraints. Formulated as an Ising model with terms enforcing both minimization of color swaps and hard constraints for each "ensemble" of cars (Yarkoni et al., 2021). Dense constraints yield fully connected subgraphs, limiting scalability for direct quantum annealing.
• Quality-Integrated Sequencing: Machine learning models predict defect probabilities conditional on sequence context, integrating expected repair costs into the objective function along with color changeover costs. The quantum cost Hamiltonian is thus adjusted to reflect both swap minimization and repair probability (Huang et al., 2022).

For large-scale industrial instances, hybrid quantum-classical solvers currently offer the best performance, with direct quantum approaches well suited to small-to-intermediate problem sizes.

7. Open Directions and Current Challenges

The inapproximability of BPSP by classical means, coupled with demonstrated quantum advantages at moderate circuit depth, positions it as an important benchmark for both quantum algorithm theory and quantum hardware development. Open directions identified include:

• Mitigating noise and hardware limitations: Adaptations such as recursive QAOA, reverse-causal-cone optimization, and parameter transfer methods aim to make quantum optimization viable for larger, noisier devices (Mooney et al., 15 Jul 2025).
• Generalizing to multi-color paint shop problems: Extensions to more than two colors are an open route for future investigation (Streif et al., 2020).
• Scaling and expressivity of quantum ansatzes: The robust performance of XQAOA—even at $p=1$—demonstrates the impact of ansatz design on practical quantum optimization, motivating further research on ansatz expressivity versus depth (Vijendran et al., 18 Sep 2025).
• Hybrid and machine learning augmentations: Integrating predictive models for process quality with combinatorial optimization, as in the coupling of repair cost and defect probability (Huang et al., 2022), represents a leading paradigm for complex scheduling and sequencing tasks in manufacturing.

The reduction to weighted MaxCut and the Ising spin model embeds BPSP within a class of problems where quantum algorithms have theoretical and practical potential to surpass classical methods, particularly as quantum devices and variational algorithm design advance.