💻 computer science

A general optimization solver based on OP-to-MaxSAT reduction

The paper proposes GORED, a general optimization solver that enhances problem-solving versatility by automatically reducing diverse optimization problems into MaxSAT instances, allowing a single unified framework to achieve solution quality comparable to specialized algorithms.

Original authors: Yuxin Zhao, Han Huang, Zhifeng Hao

Published 2026-04-27

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CC BY 4.0

Original authors: Yuxin Zhao, Han Huang, Zhifeng Hao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The "Universal Translator" for Math Problems: An Easy Guide to GORED

Imagine you are a world traveler. You go to France, and you need a French translator. You go to Japan, and you need a Japanese translator. You go to Brazil, and you need a Portuguese translator. This is how most computer programs work today when solving math problems: if you have a "Transportation Problem," you build a "Transportation Solver." If you have a "Factory Scheduling Problem," you build a "Scheduling Solver."

Building a new "translator" (or solver) for every single type of problem is exhausting, expensive, and slow.

This paper introduces a breakthrough called GORED. Instead of building thousands of different translators, the researchers built one "Universal Translator" that turns every math problem into a single, common language that one super-smart machine can understand.

1. The Problem: The "Specialist" Trap

Currently, math solvers are like specialized tools in a toolbox. A hammer is great for nails, but it’s terrible for screws. A screwdriver is great for screws, but useless for nails.

In the world of optimization (finding the "best" way to do something), we have:

The Mathematicians (Mathematical Programming): They are incredibly precise, like a high-end surgical scalpel, but they only work if the problem is shaped in a very specific, "perfect" way. If the problem gets a little messy or "nonlinear," the scalpel breaks.
The Guessers (Heuristics): They are like a Swiss Army knife. They are flexible and can handle messy problems, but they aren't always accurate. They often settle for a "good enough" answer rather than the absolute best one.

2. The Solution: The "LEGO" Method (OP-to-MaxSAT)

The researchers created a process called OP-to-MaxSAT reduction.

Think of it like this: Imagine you have a collection of complex, weirdly shaped sculptures made of clay (your math problems). You want to store them all in a standard shipping container, but they don't fit.

The researchers' "reduction" method is like a machine that takes those clay sculptures, breaks them down into tiny, standard LEGO bricks, and rebuilds them inside the container.

The LEGO bricks are called MaxSAT (a very powerful, logical language).
No matter how weird the original sculpture was—whether it was a curvy vase (nonlinear) or a jagged mountain (combinatorial)—it is now just a pile of standard bricks.

Because everything is now made of the same "bricks," you only need one single, incredibly powerful robot (the MaxSAT Solver) to put them back together and find the best possible shape.

3. How does it work? (The Three Steps)

The Blueprint (Unified Modeling): They created a way to write any math problem using a standard "language" (similar to how scientists use LaTeX) so the computer can read it easily.
The Breakdown (Reduction): The computer looks at the math rules (like "this must be less than 10") and translates them into logical "If/Then" statements (the LEGO bricks).
The Rebuilder (GORED): The solver takes those logical bricks, finds the combination that satisfies all the rules, and then translates the answer back into a human-readable number.

4. Does it actually work?

The researchers put GORED to the test against the "specialists." They threw 136 different problems at it—ranging from how to route delivery trucks to how to schedule factory machines.

The results were impressive:

It’s a Generalist: Unlike the specialists, GORED didn't need a human to "re-tune" it for different problems. It just worked.
It’s a Winner: In almost every case, GORED found the exact same perfect answer as the expensive, specialized mathematical solvers. It didn't just "guess" a good answer; it found the best one.

The Big Picture

This paper shifts the goalpost. Instead of scientists spending years designing a new algorithm for every new problem that pops up in the world, they can now focus on making one single algorithm better.

If we make the "Universal Translator" faster, every single field—from economics to engineering to biology—gets a boost instantly. It’s like upgrading the internet: you don't just make one website faster; you make the whole world faster.

Technical Summary: A General Optimization Solver Based on OP-to-MaxSAT Reduction (GORED)

1. Problem Statement

Optimization problems (OPs) are essential across engineering, economics, and scientific computing. However, current optimization methodologies suffer from a lack of generality:

Mathematical Programming Methods (e.g., Linear/Integer Programming) are theoretically rigorous but require strict mathematical structures (linearity, convexity) and manual reformulation for non-standard constraints.
Heuristic Methods (e.g., Genetic Algorithms, Particle Swarm Optimization) offer flexibility but require problem-specific operator design, lack optimality guarantees, and are prone to local optima.

The core challenge is the high degree of manual intervention required to adapt algorithms to diverse problem types, which hinders the development of a truly universal solver.

2. Methodology

The authors propose GORED, a general optimization solver that shifts the paradigm from designing specialized algorithms to using a single, unified algorithm. The methodology relies on an automated OP-to-MaxSAT reduction process.

A. Unified Modeling Language
To enable automation, the authors designed a modeling language based on LaTeX mathematical formats. This language can represent:

Objective Functions: Min/Max directions with complex arithmetic expressions.
Constraints: Equality, inequality, and domain constraints (integer, real, or enumerated sets).
Quantifiers: Universal quantifiers ( $\forall$ ) to define the scope of variables.

B. The Reduction Process (OP-to-MaxSAT)
The reduction transforms an optimization problem into a Maximum Satisfiability (MaxSAT) instance through four key steps:

Variable Encoding: Numerical variables (integer and real) are encoded into Boolean variables using signed binary fixed-point representation. This allows for controlled precision by adjusting the number of fractional bits.
Constraint Reduction: Constraints are represented as operation trees. Each node in the tree (arithmetic, relational, or domain operations) is converted into Conjunctive Normal Form (CNF) using predefined reduction rules (e.g., Full Adders, Multipliers, and Comparators). These CNFs are treated as hard clauses in MaxSAT.
Objective Reduction: The objective function is also converted into CNF hard clauses. To handle minimization, the authors use a negation/complement transformation to convert it into a maximization problem. The objective is then encoded into soft clauses with weights corresponding to the binary significance of the bits.
Complexity: The authors prove that this reduction process runs in polynomial time, ensuring that the reduction itself does not become a computational bottleneck.

C. Solving Engine
GORED uses state-of-the-art MaxSAT solvers (specifically Pacose24 in their experiments) to find the optimal Boolean assignment, which is then mapped back to the original numerical space.

3. Key Contributions

Automated Reduction Algorithm: A novel method that converts diverse optimization problems (combinatorial, numerical, linear, nonlinear) into MaxSAT instances without manual intervention.
GORED Solver: A general-purpose framework that unifies the solving of multiple problem types under a single algorithmic paradigm.
Theoretical Proof: Rigorous analysis demonstrating the polynomial time complexity of the reduction process.
Unified Modeling: A LaTeX-based grammar that bridges the gap between mathematical modeling and Boolean logic.

4. Experimental Results

The authors validated GORED using 136 instances across 11 types of optimization problems (including TSP, VRP, Job-Shop Scheduling, and Nonlinear functions like Rosenbrock).

Generality: Unlike CPLEX or Gurobi, which require different mathematical approaches for different problems, GORED uses the same MaxSAT-based logic for all.
Solution Quality: GORED achieved optimal or near-optimal solutions comparable to specialized solvers (CPLEX, Gurobi) and outperformed many heuristic methods (GA, EA, PSO) in complex non-convex landscapes.
Precision vs. Error: Experiments showed that numerical errors decrease exponentially as the number of fractional bits (precision) increases, proving the reliability of the fixed-point encoding.

5. Significance

This work represents a significant shift in optimization research. By moving away from "problem-specific algorithm design" toward "automated problem reduction," GORED enables:

Cross-disciplinary Integration: A single improvement in MaxSAT solver technology can immediately benefit all optimization domains (transportation, manufacturing, etc.).
Reduced Human Effort: It eliminates the need for experts to manually reformulate non-linear or non-standard problems into standard mathematical forms.
A New Paradigm: It establishes a framework where the complexity of the problem is handled by the reduction logic, while the complexity of the search is handled by a single, highly optimized Boolean engine.