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Implementation of Polynomial NP-Complete Algorithms Based on the NP Verifier Simulation Framework

Changryeol Lee

TL;DR

The paper constructs explicit deterministic Turing machines for canonical NP-complete problems (SAT and Subset-Sum) within the Polynomial-Time NP Verifier Simulation Framework and implements a complete, executable pipeline that maps verifier computations to feasible graphs. It introduces two key innovations: a definition-faithful, reduced-edge verification strategy that tightens graph parameters (width and height) and a walk-first approach that minimizes verifier invocations, enabling a polynomial-time simulation with refined bounds such as $O(n^{16}\log n)$ for SAT and Subset-Sum instances. Beyond correctness, the work demonstrates FNP witness extraction from accepted walks, integrating certificate generation into the same polynomial-time process. The practical implementation emphasizes transparency and reproducibility, with Python code and data structures provided, while acknowledging large constants and limitations in scale, and outlining future directions toward RAM-based models and further asymptotic reductions. Together, these contributions advance the concrete realization of the NP verification framework and illuminate the trade-offs between formal guarantees and implementability in explicit-machine constructions.

Abstract

While prior work established a verifier-based polynomial time framework for NP, explicit deterministic machines for concrete NP-complete problems have remained elusive. In this paper, we construct fully specified deterministic Turing Machines (DTMs) for SAT and Subset-Sum within a polynomial-time NP verifier simulation framework. We show that both machines operate in polynomial time and, for satisfiable instances, deterministically generate valid witnesses, thereby extending the framework to deterministic FNP computation without increasing the degree of polynomial complexity. Furthermore, we provide a complete implementation of the framework, including the dynamic computation graph, feasible-graph construction, verification walks, and Turing-machine simulation via edge extensions. The implementation behaves in accordance with the predicted polynomial-time bounds. To ensure transparency and reproducibility, the complete Python implementation and source code are made available in a public online repository.

Implementation of Polynomial NP-Complete Algorithms Based on the NP Verifier Simulation Framework

TL;DR

The paper constructs explicit deterministic Turing machines for canonical NP-complete problems (SAT and Subset-Sum) within the Polynomial-Time NP Verifier Simulation Framework and implements a complete, executable pipeline that maps verifier computations to feasible graphs. It introduces two key innovations: a definition-faithful, reduced-edge verification strategy that tightens graph parameters (width and height) and a walk-first approach that minimizes verifier invocations, enabling a polynomial-time simulation with refined bounds such as for SAT and Subset-Sum instances. Beyond correctness, the work demonstrates FNP witness extraction from accepted walks, integrating certificate generation into the same polynomial-time process. The practical implementation emphasizes transparency and reproducibility, with Python code and data structures provided, while acknowledging large constants and limitations in scale, and outlining future directions toward RAM-based models and further asymptotic reductions. Together, these contributions advance the concrete realization of the NP verification framework and illuminate the trade-offs between formal guarantees and implementability in explicit-machine constructions.

Abstract

While prior work established a verifier-based polynomial time framework for NP, explicit deterministic machines for concrete NP-complete problems have remained elusive. In this paper, we construct fully specified deterministic Turing Machines (DTMs) for SAT and Subset-Sum within a polynomial-time NP verifier simulation framework. We show that both machines operate in polynomial time and, for satisfiable instances, deterministically generate valid witnesses, thereby extending the framework to deterministic FNP computation without increasing the degree of polynomial complexity. Furthermore, we provide a complete implementation of the framework, including the dynamic computation graph, feasible-graph construction, verification walks, and Turing-machine simulation via edge extensions. The implementation behaves in accordance with the predicted polynomial-time bounds. To ensure transparency and reproducibility, the complete Python implementation and source code are made available in a public online repository.
Paper Structure (118 sections, 31 theorems, 90 equations, 1 figure, 13 tables, 28 algorithms)

This paper contains 118 sections, 31 theorems, 90 equations, 1 figure, 13 tables, 28 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background and Preliminaries
  4. Deterministic Turing Machines and Computation Graph
  5. Feasible Graph
  6. Overview of Polynomial-Time NP Verifier Simulation Framework
  7. Summary of Time Complexity Bounds (Original Framework).
  8. Explicit Turing Machines for NP-Complete Problems
  9. Remark.
  10. SAT Turing Machine with Input-Dependent States
  11. High-Level Structure
  12. Transition-Level Structure
  13. Transition Table (High-Level, Parameterized States)
  14. Bounded Parameterized States
  15. Discussion
  16. ...and 103 more sections

Key Result

Lemma 1

If $\mathcal{M}_{\mathrm{SAT}}^{\mathrm{ID}}$ halts in the Accept state on input $\Phi \# \mathcal{V}$, then the valuation $\mathcal{V}$ satisfies the CNF formula $\Phi$.

Figures (1)

  • Figure 1: Types of Combining edges

Theorems & Definitions (113)

  • Definition 1: Deterministic Single-Tape Turing Machine
  • Definition 2: Acceptors and Verifiers
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9: Previous and Next Edges
  • Definition 10
  • ...and 103 more