[Paper Review] A Tutorial on Formulating QUBO Models.

Fred Glover, Gary Kochenberger|arXiv (Cornell University)|Nov 13, 2018

Operations Management Techniques20 citations

TL;DR

This paper presents a systematic tutorial on formulating combinatorial optimization problems as Quadratic Unconstrained Binary Optimization (QUBO) models using straightforward reformulation techniques. By transforming classical optimization problems into QUBO form, the method enables efficient solution via specialized QUBO solvers, establishing QUBO as a unifying framework for diverse industrial and governmental optimization challenges.

ABSTRACT

The field of Combinatorial Optimization (CO) is one of the most important areas in the general field of optimization, with important applications found in every industry, including both the private and public sectors. It is also one of the most active research areas pursued by the research communities of Operations Research, Computer Science, and Analytics as they work to design and test new methods for solving real world CO problems. In recent years, we have discovered that a mathematical formulation known as QUBO, an acronym for a Quadratic Unconstrained Binary Optimization problem, can embrace a large variety of important CO problems found in industry and government. Through special reformulation techniques that are easy to apply, the power of QUBO solvers can be used to efficiently solve many important problems once they are put into the QUBO framework. This two-step process of first re-casting an original model into the form of a QUBO model and then solving it with appropriate software enables the QUBO model to become a unifying framework for combinatorial optimization. The alternative path that results for effectively modeling and solving many important problems is a new development in the field of combinatorial optimization. The materials provided in subsequent sections illustrate the process of reformulating important optimization problems as QUBO models through a series of explicit examples. Collectively these examples highlight the application breadth of the QUBO model. We do not focus here on modeling a given problem in the form classically adopted in the optimization community. Rather, our examples disclose the methods that can be used to recast a given problem, perhaps existing in a classical mathematical form, into an equivalent QUBO model.

Motivation & Objective

• To demonstrate that a wide range of combinatorial optimization problems can be reformulated into QUBO form.
• To provide a unified modeling framework that enables the use of QUBO solvers for diverse real-world problems.
• To shift focus from classical optimization modeling to QUBO reformulation as a powerful alternative approach.
• To illustrate the broad applicability of QUBO through explicit, step-by-step examples.
• To empower researchers and practitioners with tools to transform existing models into QUBO-compatible forms.

Proposed method

• The paper employs reformulation techniques to convert classical mathematical models of combinatorial optimization problems into equivalent QUBO formulations.
• Binary variables are introduced to represent discrete choices, enabling the expression of logical and combinatorial constraints in quadratic form.
• Constraints are embedded into the objective function using penalty terms to maintain feasibility in the QUBO formulation.
• The method ensures equivalence between the original problem and the resulting QUBO model through mathematical transformation.
• Examples are used to demonstrate the process, showing how different problem types (e.g., set partitioning, graph partitioning) can be mapped to QUBO.
• The approach avoids reliance on traditional optimization modeling formats, focusing instead on QUBO-specific reformulation.

Experimental results

Research questions

• RQ1 How can classical combinatorial optimization problems be systematically transformed into QUBO models?
• RQ2 What reformulation techniques are effective for converting constraints and objectives into quadratic binary forms?
• RQ3 To what extent can QUBO serve as a unifying framework across diverse optimization problems?
• RQ4 What are the key steps and principles for ensuring equivalence between original models and their QUBO reformulations?
• RQ5 How can practitioners apply QUBO modeling to real-world problems without prior expertise in quantum-inspired optimization?

Key findings

• The QUBO framework can represent a broad spectrum of combinatorial optimization problems, including those in industry and government.
• Reformulation into QUBO form enables efficient solution using specialized QUBO solvers, improving problem-solving scalability.
• The transformation process is systematic and accessible, relying on well-defined techniques applicable to various problem types.
• The method preserves problem equivalence, ensuring that solutions to the QUBO model are valid for the original problem.
• The tutorial demonstrates that QUBO modeling is a viable and powerful alternative to classical optimization modeling approaches.
• The approach highlights the potential of QUBO as a unifying paradigm in combinatorial optimization.