OPS Benchmark: One-to-Many Problem-Solution

• OPS Benchmark is a framework that formalizes one-to-many mapping challenges across voice conversion, matching markets, and optimization scenarios.
• It employs domain-specific techniques such as dynamic frequency warping, synthetic preference matching, and tunable GPD generators to craft diverse problem instances.
• Metrics including spectral fidelity, assignment stability, and Pareto front characterization provide actionable insights into model performance and robustness.

The One-to-Many Problem-Solution (OPS) Benchmark formalizes and evaluates the ability of models and algorithms to address problem domains in which a single problem setup admits multiple, structurally distinct solutions. These benchmarks isolate phenomena arising in real-world settings—predominantly in voice conversion, combinatorial optimization, and multi-objective optimization—where mapping uniqueness, stability, and optimality cannot be trivially guaranteed. The OPS benchmark concept provides structured testbeds for model development, comparative evaluation, and methodological innovation across statistical mapping, LLM reasoning, and many-objective optimization contexts.

1. The One-to-Many Mapping Phenomenon

The one-to-many mapping problem emerges when a source instance admits multiple valid targets due to underlying variability, context-dependence, or preference constraints. This violates the assumption of functional determinism found in many standard modeling scenarios. In statistical voice conversion (VC), the mapping $M:X \to Y$ between source and target feature domains is often learned under parallel data and minimum mean squared error (MMSE) loss. However, nearly identical input vectors $x_i \approx x_j$ can correspond to highly dissimilar outputs $y_i, y_j$ due to factors such as rendition variability or formant mismatches (Mohammadi, 2015). In matching markets, the canonical College Admission Problem instantiates one-to-many structure through agent preferences and institutional capacity constraints, rendering the solution space combinatorially rich and stability-sensitive (Fauchard et al., 16 Sep 2025). In many-objective optimization, mapping from a high-dimensional feasible space to the Pareto front naturally generates many valid optima for a single trade-off scenario (Meneghini et al., 2020). The OPS benchmark unifies these scenarios by formalizing them as settings with non-functional or multi-valued mappings, specifically designed to stress solution quality, robustness, and the ability to respect nuanced constraints.

2. OPS Benchmark Designs: Domains and Construction

Voice Conversion (Formant Equalization)

In voice conversion, the OPS benchmark targets the "one-to-many" effect by constructing source-target frame pairs where the spectral mapping cannot be captured by a deterministic function. The pipeline comprises: (1) frame-wise formant extraction via LPC-based methods, (2) frame alignment using Dynamic Time Warping (DTW), (3) formant location equalization using per-frame Dynamic Frequency Warping (DFW), and (4) GMM-based training on the equalized features, followed by inverse DFW to restore the natural formant structure post-conversion (Mohammadi, 2015).

Preference-Based Matching (College Admissions)

The matching market OPS benchmark is instantiated via synthetic generations of the College Admission Problem: multiple students (agents) compete for college placements with both sides expressing strict preferences and capacity limits (Fauchard et al., 16 Sep 2025). Benchmark instances vary the number of agents, preference completeness, total capacity (under/over/exact), and student-college ratios, yielding a diverse suite of 369 evaluation problems. Evaluation rigorously checks feasibility (capacity and assignment), assignment stability (no "invalid" matches), matching stability (no blocking pairs), and student-optimality in the lattice of stable matchings.

Many-Objective Optimization

The OPS benchmark in many-objective optimization is systematically constructed via the Generalized Position–Distance (GPD) tunable generator (Meneghini et al., 2020). Decision variables are split into position and distance (to front) components, with objectives formed multiplicatively or additively. Parameter sweeps control the number of objectives, front shape (via a $p$-norm), multimodality, robustness, bias, and domain constraints (e.g., φ-inequalities and ξ-equalities), enabling the programmable creation of infinitely many OPS test problems with analytically known Pareto sets/fronts.

Domain	Mapping Structure	Benchmark Construction
Voice Conv.	Source $\to$ many targets	Formant-aligned DFW, GMM regression
Matching	Applicants $\to$ colleges	Preference/Capacity–based synthesis
Optimization	$\mathbb{R}^N \to$ Pareto	GPD generator, multimodal fronts

3. Evaluation Criteria and Metrics

OPS benchmarks employ domain-specific and general metrics that target the major pathologies and strengths exposed by one-to-many mapping:

• Complexity/Functionality: In VC, the weighted covariance determinant of mapped outputs conditioned on input ($\det(\mathrm{WeightedCov}_{x'})$) quantifies mapping complexity pre- and post-formant equalization (Mohammadi, 2015).
• Distortion/Quality: Mel-cepstral distortion (melCD) measures spectral fidelity in VC; subjective crowdsourced CMOS tests assess perceived speech quality.
• Feasibility and Stability: In matching benchmarks, solution outputs are automatically checked for assignment feasibility, pairwise blocking conditions (stability), and optimality against agent preferences (Fauchard et al., 16 Sep 2025).
• Optimality: Student-optimality is formally defined as the matching achieving minimum sum of student ranks—computed by Deferred Acceptance in $O(|S||C|)$ time.
• Front Characterization: Analytical derivation of the Pareto front in GPD-based benchmarks, with closed-form geometric characterizations, allows precise assessment of approximation quality and solution diversity (Meneghini et al., 2020).

Additional metrics cover output validity (syntactic/format correctness for LLMs), average inference runtime, and sensitivity to domain parameters (market size, preference structure, objective count).

4. Methodological Approaches and Prompting Strategies

Benchmark solutions leverage a spectrum of model families and prompting protocols, with design closely coupled to the problem's one-to-many nature:

• Voice Conversion: Gaussian mixture models (GMMs) are standard for regression but, when trained on formant-equalized spectra, achieve simplification of the regression landscape. The transformation pipeline benefits from explicit DFW for formant alignment and the potential for extension to neural mappers after variability removal (Mohammadi, 2015).
• LLM Reasoning in Matching: The matching OPS benchmark systematically compares prompt styles: basic descriptions, role-based setups, in-context learning (ICL) with/without stepwise execution, Chain-of-Thought (CoT) in natural language, pseudocode, and Python. Iterative prompting with automated feedback (indicating which criteria failed) samples output variation but does not ensure monotonic improvement (Fauchard et al., 16 Sep 2025).
• Optimization Algorithms: For many-objective cases, meta-heuristics and evolutionary algorithms are evaluated using the mathematically tractable GPD-generated problems. Parameter choices grant rigorous control over multimodality and deceptive/robust features (Meneghini et al., 2020).

A key finding is that "reasoning-tuned" LLMs (e.g., QwQ 32B, GPT-oss 120B) substantially outperform base models on feasibility and stability, but increased market size or front complexity sharply reduce performance. Prompting templates need to be matched to model class, with base models benefitting from structured, explicit guides and reasoning-tuned models better served by lighter scaffolding.

5. Empirical Findings and Comparative Results

Benchmarks across domains exhibit distinctive empirical signatures:

• Voice Conversion: Formant normalization by DFW reduces mapping complexity (visualized by lower covariance and simpler PCA projections) and significantly lowers melCD (from 9.23 dB to 8.38 dB) (Mohammadi, 2015). CMOS listening tests demonstrate statistically significant preference for DFW-equalized systems (p < 0.01), with no loss of speaker similarity.
• Matching Markets: Reasoning-tuned LLMs achieve >98% feasibility and ≈80% stability on small markets, but performance is non-monotonic under iterative prompting, and degrades with increasing |S|. No single prompt format dominates; open questions remain regarding feedback efficacy (Fauchard et al., 16 Sep 2025).
• Optimization: GPD-based instances allow complete analytical tractability of the Pareto front, and parameterized generation exposes algorithm weaknesses by introducing bias, multimodality, and deceptive features that stress evolutionary and hybrid solvers (Meneghini et al., 2020).

6. Customization, Scaling, and Automation Guidelines

OPS benchmarks’ modular construction enables controlled scaling:

• Objective Count & Front Geometry: Choose $M$ and the $p$-norm to determine convexity/concavity in optimization (Meneghini et al., 2020).
• Distance/Bias Mechanics: Manipulate $q, t, S$ to generate interactions in position and complexity in distance-to-front variables.
• Problem Difficulty: Adjust market size (|S|,|C|), preference completeness, or parameterized multimodality/robustness to tune computational and reasoning challenge.
• Automated Instance Generation: For GPD, comprehensive test suites are generated by systematically sweeping parameters (M, p, q, t, k, S, constraint forms), enabling large-scale benchmarking and reproducible evaluation.
• Domain Adaptation: Extension to new tasks is feasible by abstracting OPS core principles—multi-valued mapping, solution diversity, structural constraints—and synthesizing or adapting problem instances that maximize discriminatory power for targeted solution methods.

7. Limitations and Prospective Directions

Known limitations surface from both underlying problem structure and model capabilities:

• Formant-based DFW: Susceptible to errors in automatic formant tracking and bandwidth estimation, leading to potential spectral artifacts or suboptimal warping (Mohammadi, 2015).
• LLM Matching Reasoning: Capacity does not substitute for specialized tuning; iterative prompting does not guarantee improvement, and error-correcting feedback lacks reliability for large instances (Fauchard et al., 16 Sep 2025).
• Optimization Benchmarking: While the GPD generator offers infinite test problems with known solutions, real-world many-objective tasks often present further structural subtleties (e.g., physical constraints, stochasticity) not perfectly captured by parametric design (Meneghini et al., 2020).

Planned extensions include deploying hand-corrected or ground-truth labels for isolating the upper bound of normalization effects, integrating DNN-based regression after formant equalization in VC, richer instance-specific feedback mechanisms for LLMs, and expanding the GPD library to model problem classes beyond those addressed in current parameterizations.

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References:

• "Reducing one-to-many problem in Voice Conversion by equalizing the formant locations using dynamic frequency warping" (Mohammadi, 2015)
• "Reasoning with Preference Constraints: A Benchmark for LLMs in Many-to-One Matching Markets" (Fauchard et al., 16 Sep 2025)
• "Scalable and Customizable Benchmark Problems for Many-Objective Optimization" (Meneghini et al., 2020)