NitroSAT

A High-Performance O(M) MaxSAT approximator with unusually good scaling behavior and high satisfaction rates on massive instances. Using Physics-Informed Continuous Relaxation. Made in India by Shunyabar Labs.

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Overview

NitroSAT is a high-performance MaxSAT approximator with unusually good scaling behavior and high satisfaction rates on massive instances that achieves O(M) linear time complexity relative to the number of clauses. Unlike traditional CDCL-based solvers, NitroSAT treats Boolean satisfiability as a physics-informed dynamical system on a Riemannian manifold, using continuous relaxation, spectral methods, and topological analysis. Find the code here.

We threw thousands of problems at NitroSAT—everything from domset_4 to 7-million-clause planted coloring monsters to random 3SAT, 5,000+ CNF instances and solver came out with a 77% perfect SATISFIED rate and a 99.7% median satisfaction. Assignments: https://huggingface.co/datasets/sethuiyer/navokoj_sat_2024 and benchmarks/README.md.

Note from GPT-5.5: NitroSAT v2 looks very strong as a fast approximate/heuristic satisfiability engine, especially for high-satisfaction or MaxSAT-style use, but the formal proof and benchmark hygiene need tightening before making stronger solver-theory claims.

Some notable results:

Test Name / Problem Type	Variables	Clauses	Ratio ($\alpha$)	Latency	Satisfaction
Planted Coloring (Lua)	105,000	232,043	2.21	13.78s	100%
Grid Coloring (1000x1000)	4,000,000	14,992,000	3.75	475.0s	100%
Massive Hardware Check	788,480	2,617,349	3.32	5.92s	100%
Enterprise Timetabling	147,600	80,278,884	543.89	73sec	100%
Titan Ramsey Density	780	1,316,016	1,687.20	3,403.0s	99.995%
Adversarial Pitfall Trap	2,950	1,047,620	355.12	~400.0s	100%
Large Phase Transition	200,000	852,000	4.26	11.2 min	99.20%
Ramsey R(5,5,5)	~4,760	354,890	74.56	3.5s	100%
Small-World Lattice	360,000	1,354,800	3.76	62.75s	100%

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Key Features

• Linear Scaling — O(M) time complexity relative to clause count
• Structural Awareness — Detects UNSAT via thermodynamic phase transitions
• Zero Configuration — Works out-of-the-box on Scheduling, Ramsey, Coloring, N-Queens
• Multi-Phase Architecture — Langevin flow, topological repair, and adelic saturation phases
• UNSAT Detection — GF(2) parity checking and topological probe ordering
• No External Dependencies — Standalone C99 implementation
• Lua Implementation — LuaJIT version available via Codeberg

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Algorithm Overview

NitroSAT maps the Boolean satisfiability problem to an energy landscape, using spectral initialization and Branch-Aware Holonomy Annealing (BAHA) to navigate complex basins.

1. Continuous Relaxation — Maps Boolean variables to continuous values in [0,1]
2. Prime Weighting — Number-theoretic clause weights based on the Prime Number Theorem
3. Heat Kernel Diffusion — Spectral smoothing via the heat kernel exp(t * Δ)
4. BAHA (Branch-Aware Holonomy Annealing) — Uses Lambert-W function for phase transitions
5. Persistent Homology — Tracks topological holes (Betti numbers) to guide repair
6. NADAM Optimization — Nesterov-accelerated adaptive moment estimation

See MATH.md for the complete mathematical theory.

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Quick Start

1. Compile

gcc -O3 -march=native -std=c99 nitrosat.c -o nitrosat -lm

Requirements: GCC or any C99-compatible compiler + standard math library (-lm)

2. Run

./nitrosat problem.cnf

JSON is the default output format. Progress milestones are printed to stderr so stdout stays clean for piping:

NitroSAT  problem.cnf  | 625 vars  24825 clauses
[pass 1/5]   [stagnation] 99.99% plateau → jumping to finisher
  [phase-2] topological repair... done → 24823/24825 (99.99%)
  [phase-3] adelic saturation... done → 24820/24825 (99.99%)

3. Command-Line Options

Option	Description
<cnf-file>	Input file in DIMACS CNF format
[max-steps]	Maximum optimization steps (default: adaptive)
--no-dcw	Disable prime-weighted clause learning
--no-topo	Disable topological repair phase
--json	Output JSON format (default)
--cinematic	Verbose per-step logging
--proof <path>	Generate DRAT proof for UNSAT instances
--proof-format drat	Proof format (default: drat)

4. Piping & Integration

# Extract the assignment
./nitrosat problem.cnf | jq '.assignment'

# Check if solved
./nitrosat problem.cnf | jq '.status'

# Save result to file (progress still prints to terminal)
./nitrosat problem.cnf > result.json

# Parse in Python
./nitrosat problem.cnf | python3 -c "import json,sys; d=json.load(sys.stdin); print(d['satisfaction_rate'])"

5. Cinematic Mode (Verbose)

./nitrosat problem.cnf --cinematic

6. UNSAT Certificate Mode (DRAT)

./nitrosat problem.cnf --proof proof.drat --proof-format drat --json

The proof backend is solver-aware — it uses NitroSAT's own UNSAT detection signals to guide proof generation:

1. UNSAT core extraction — extracts the irreducible core after solving
2. GF(2) parity check — detects odd-cycle contradictions in the binary clause implication graph
3. Topological probe ordering — failed-literal probing ordered by prime-weighted irreducibility pressure
4. Solver diagnostics — reports UNSAT awareness signals: unsatisfied count, fractures, beta1, convexity

Proof status values in JSON output:

• generated_top_level_up_conflict — contradiction found by unit propagation
• generated_gf2_parity_refutation — contradiction from odd parity cycle
• generated_topo_failed_literal_refutation — contradiction via math-guided probing
• inconclusive — no proof derived; signals explain why

Note: The backend is exact (sound) but incomplete. Pigeonhole-style instances require extended resolution for short proofs.

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JSON Output Format

{
  "status": "SATISFIED",
  "satisfaction_rate": 1.0,
  "satisfied": 425,
  "unsatisfied": 0,
  "variables": 150,
  "clauses": 425,
  "assignment": [-1, -2, 3, -4, -5, 6],
  "latency": {
    "total_ms": 54.12,
    "throughput": "7.9K clauses/sec"
  }
}

Key fields:

Field	Description
confidence	Raw continuous values (0.0–1.0) before rounding to Boolean
critical_beta	Predicted phase transition point from the Prime Number Theorem
convexity_status	STABLE (convex regime) or NON_CONVEX (glassy phase)
betti_0 / betti_1	Topological invariants of the unsatisfied clause complex
blame	Top-20 unsatisfied clauses ranked by prime weight (UNSAT debugging)
proof	UNSAT certificate request/result metadata

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Citation

If you use NitroSAT in your research, please cite:

@software{sethurathienam_iyer_2026_18753235,
  author       = {Sethurathienam Iyer},
  title        = {NitroSAT: A Physics-Informed MaxSAT Solver},
  year         = 2026,
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.18753235},
  url          = {https://doi.org/10.5281/zenodo.18753235}
}

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Reproducibility

The CNF files used to test NitroSAT are available at:

• HuggingFace Dataset

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Implementations

Implementation	Location
NitroSAT (C)	GitHub · Zenodo
NitroSAT (Lua)	Codeberg
Navokoj API	navokoj.shunyabar.foo

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License

NitroSAT is licensed under the Apache License 2.0. See the LICENSE file for details.

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Support & Contributing

• Issues: Report bugs and feature requests via GitHub Issues
• Email: Contact the maintainer at shunyabarlabs@zohomail.com
• ORCID: 0009-0008-5446-2856
• Sponsor: Support development at github.com/sponsors/sethuiyer

Contributions are welcome. Please follow the Code of Conduct.