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Journal of Information and Optimization Sciences

QUBO formulations of three NP problems

Anuradha Mahasinghe, Vishmi Fernando, Paduma Samarawickrama

Published Oct 3, 2021• 6 citations
Publisher

Abstract

With the recent advancements in the field of combinatorial optimization, QUBO (quadratic unconstrained binary optimization) has become a topic of intense interest, as a problem format acceptable to several hardware heuristics such as quantum and digital annealers. Accordingly, transforming hard mathematical problems into QUBO problems is of particular importance. In this paper, we restate three problems in NP-complete and NP-hard classes as QUBO problems with proofs of equivalence. Our problems of interest are the numerical three-dimensional matching problem, the monochromatic triangle problem and the optimum facility location problem. Furthermore, we comment on the feasibility of these formulations on sophisticated QUBO solving platforms.

DOI
10.1080/02522667.2021.1930657

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•Nov 13, 2018•24 citations
With the recent advancements in the field of combinatorial optimization, QUBO (quadratic unconstrained binary optimization) has become a topic of intense interest, as a problem format acceptable to several hardware heuristics such as quantum and digital annealers. Accordingly, transforming hard mathematical problems into QUBO problems is of particular importance. In this paper, we restate three problems in NP-complete and NP-hard classes as QUBO problems with proofs of equivalence. Our problems of interest are the numerical three-dimensional matching problem, the monochromatic triangle problem and the optimum facility location problem. Furthermore, we comment on the feasibility of these formulations on sophisticated QUBO solving platforms.
Quadratic Unconstrained Binary Optimization (QUBO) models are defined by quadratic expressions composed of binary variables, each taking a value of either 0 or 1. The objective of the QUBO problem is to find an optimal assignment of these binary variables that minimizes the value of the QUBO model. The QUBO problem has attracted significant research interest due to its versatility in reformulating various NP-complete problems and its potential for efficiently finding heuristic solutions through quantum annealing. This paper focuses on converting Cubic Unconstrained Binary Optimization (CUBO) problems, defined by cubic expressions, into equivalent QUBO models. Through this conversion, CUBO problems can be solved using quantum annealing. Since the important NP-hard problem 3-SAT can be straightforwardly represented as a CUBO problem, investigating the conversion from CUBO to QUBO models is a meaningful endeavor.
Recent advances in quantum computing and the increasing availability of quantum hardware have substantially enhanced the practical relevance of quantum approaches to discrete optimization. Among these, the Quadratic Unconstrained Binary Optimization (QUBO) formulation provides a unifying modeling framework for a broad class of NP\mathbf{NP}NP-hard problems and is naturally suited to quantum computing and quantum-inspired algorithms. Location science, network design, and logistics represent core application domains of discrete optimization, combining high practical impact with substantial computational challenges. In this work, we develop QUBO formulations for several fundamental problems in these domains, including a nonlinear integer formulation of the Discrete Ordered Median Problem (DOMP). Beyond their modeling relevance, these QUBO formulations serve as representative benchmark problems for assessing quantum algorithms and quantum hardware. We further derive a tight bound for the penalty parameter ensuring equivalence between the QUBO formulation and its underlying integer program. Finally, we conduct a comprehensive computational study using QAOA, WS-QAOA, and classical heuristics for QUBO instances of the ppp-Median Problem and the Fixed-Charge Facility Location Problem (FCFLP), and introduce two effective warm-start strategies for WS-QAOA based on its linear programming relaxation.
Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The graph isomorphism problem is one of the applications of combinatorial optimization, and is not known to be solvable in polynomial time. The problem is also not known to be NP-complete. We propose four QUBO formulations for graph isomorphism problem instances. We also propose four QUBO formulations for induced subgraph isomorphism problem instances and two QUBO formulations for subgraph isomorphism problem instances, which are NP-complete problems. We solve QUBO instances defined by our QUBO formulations, using known QUBO solvers: Gurobi optimizer, Fixstars Amplify AE and OpenJij with SA.
Quadratic Unconstrained Binary Optimization (QUBO) is a central formulation for combinatorial optimization and has gained increasing attention due to its compatibility with quantum, hybrid quantum-classical, and quantum-inspired solvers. However, translating natural-language problem descriptions into correct QUBO formulations remains difficult, requiring the identification of binary variables, constraints, objective functions, penalty terms, and suitable penalty weights. This process is time-consuming and often demands substantial domain expertise. To address this challenge, we propose an end-to-end multi-agent framework that automatically generates QUBO formulations from natural-language problem descriptions, supported by structured or unstructured test cases. To evaluate its performance, We also introduce QUBOBench, a benchmark containing 100 combinatorial optimization problems across 12 application domains, curated from peer-reviewed literature, competitions, and canonical NP-hard problems. Experimental results show that our framework achieves 68% accuracy on QUBOBench, outperforming a direct single-call baseline by 22%. Further analysis identifies iterative self-repair as the most important component contributing to improved performance. The data and code are open-sourced at https://quitttcat.github.io/QuantumQUBOAgent.
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.