Position-Candidate-Hypothesis (PCH) Paradigm
A theoretical paradigm for structural-statistical analysis of NP-complete problems
Shifting focus from combinatorial search to structural decomposition.
Abstract — This research paper introduces the Position-Candidate-Hypothesis (PCH) paradigm as a novel theoretical approach to NP-complete problems. This work proposes a fundamental shift from traditional combinatorial search to structural-statistical analysis. The research explores the decomposition of problems into three interconnected components: positions, candidates, and hypotheses, followed by statistical integration.
Published: November 2025 · Status: Theoretical Research
| Detail | Information |
|---|---|
| Title | Position-Candidate-Hypothesis (PCH) Paradigm |
| DOI | 10.5281/zenodo.17614888 |
| Published | November 2025 |
| License | Creative Commons Attribution 4.0 International |
| Author | Alexander Suvorov |
| ORCID | 0009-0006-3427-9611 |
| Type | Theoretical Research |
PCH paradigm operates through structural decomposition:
Problem Analysis → Structural Decomposition → Parallel Investigation → Statistical Synthesis
| Component | Description |
|---|---|
| Positions (n) | Structural elements in solution space. For problem size n, there are n positions. |
| Candidates (n) | Entities for position assignments. Each position considers n candidates. |
| Hypotheses (n) | Independent research processes. n hypotheses provide complete problem coverage. |
Research Proposition: PCH uses n hypotheses, n positions, and n candidates per position for problems of size n.
Hypothesis Investigation:
| Hypothesis | Starting Candidate |
|---|---|
| h₁ | candidate c₁ |
| h₂ | candidate c₂ |
| ... | ... |
| hₙ | candidate cₙ |
Research Phases:
| Phase | Description |
|---|---|
| Hypothesis Launch | n independent hypotheses start |
| Position Research | All n positions examined |
| Candidate Evaluation | Potential assignments analyzed |
| Statistical Integration | Findings synthesized across hypotheses |
Parallelism:
| Level | Description |
|---|---|
| Hypothesis-level | All n hypotheses run concurrently |
| Position-level | Position research parallelized |
| Distributed | Hypotheses run on multiple nodes |
Investigation:
- Independent research trajectories
- Minimal cross-hypothesis contamination
- Simultaneous diverse solution exploration
PCH shifts problem-solving perspective:
| From | To |
|---|---|
| Search | Structure |
| Sequential | Parallel |
| Deterministic | Statistical |
| Black-box | Interpretable |
Detailed Transformations:
- Search → Structure — Architectural analysis focus
- Sequential → Parallel — Simultaneous investigation
- Deterministic → Statistical — Probabilistic synthesis
- Black-box → Interpretable — Transparent solution derivation
PCH applies to NP-complete problems:
- TSP — Traveling Salesman Problem
- SAT — Boolean Satisfiability Problem
- Knapsack — Knapsack Problem
- Graph Coloring — Graph Coloring Problem
- Vertex Cover — Vertex Cover Problem
- Hamiltonian Path — Hamiltonian Path Problem
Before the Study:
The following implementations represent practical work conducted prior to the formalization of the PCH paradigm:
- Exact TSP Solver — High-performance exact solver for TSP implemented in Go with Branch and Bound algorithm
- Smart TSP Benchmark — Professional algorithm testing infrastructure with configurable scenarios and detailed metrics
- Smart TSP Oracle — High-performance exact solver for TSP implemented in Python
- Smart TSP Solver — High-performance Python library with novel heuristic approaches outperforming classical methods by 25%
After the Study:
Research in this area is ongoing. New implementations based on the PCH paradigm are currently in development and will be released upon completion.
Note: The PCH paradigm is a theoretical framework. Practical implementations based on this paradigm are under active research and development.
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@misc{suvorov_2025_17614888,
author = {Suvorov, Alexander},
title = {Position-Candidate-Hypothesis (PCH) Paradigm: A
Structural-Statistical Approach to NP-Complete
Problems},
month = nov,
year = 2025,
publisher = {Zenodo},
doi = {10.5281/zenodo.17614888},
url = {https://doi.org/10.5281/zenodo.17614888},
}
ACADEMIC RESEARCH — NOT FOR PRACTICAL USE
This is theoretical research, not production software.
Warning: This work is for academic discussion only. Do not use in production systems.
Full academic disclaimer: See DISCLAIMER.md