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arXiv:2410.21074 · Problems of Information Transmission

Technique for Transforming Discrete Optimization Problems into QUBO Form

A. M. Semenov, Sergey R. Usmanov, Aleksey K. Fedorov

Published Sep 1, 2025• 2 citations
arXivHTMLPDF

Abstract

Practical discrete optimization problems often contain multidimensional arrays of variables interrelated by linear constraints such as equalities and inequalities. Values of each variable depend on its specific meaning and can be binary, integer, or discrete. These conditions make it technically difficult to reduce the original problem statement to QUBO form. We identify and examine three necessary transformations of the original problem statement to reduce it to QUBO form, namely transition from a multidimensional to a one-dimensional array, transition to binary variables in mixed problems, and incorporating linear constraints into the objective function in the form of quadratic penalties. We present and prove computationally convenient formulas to simplify these transformations. In particular, the formulas for the transition from a multidimensional to a one-dimensional array of variables are based on the application of the Kronecker product of matrices. The transformations considered are illustrated by numerous examples and used, as an application, to reduce a number of well-known problems in graph theory and combinatorial optimization to QUBO form.

DOI
10.1134/s0032946025020036

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Practical discrete optimization problems often contain multidimensional arrays of variables interrelated by linear constraints such as equalities and inequalities. Values of each variable depend on its specific meaning and can be binary, integer, or discrete. These conditions make it technically difficult to reduce the original problem statement to QUBO form. We identify and examine three necessary transformations of the original problem statement to reduce it to QUBO form, namely transition from a multidimensional to a one-dimensional array, transition to binary variables in mixed problems, and incorporating linear constraints into the objective function in the form of quadratic penalties. We present and prove computationally convenient formulas to simplify these transformations. In particular, the formulas for the transition from a multidimensional to a one-dimensional array of variables are based on the application of the Kronecker product of matrices. The transformations considered are illustrated by numerous examples and used, as an application, to reduce a number of well-known problems in graph theory and combinatorial optimization to QUBO form.
Practical discrete optimization problems often contain multidimensional arrays of variables with linear constraints, which complicates their conversion to QUBO (quadratic unconstrained binary optimization) form. The article proposes a systematic approach to transforming such problems, which includes three key steps: the transition from a multidimensional representation of variables to a one-dimensional one using the Kronecker product of matrices, the reduction of mixed variables to binary ones, and the introduction of linear constraints into the objective function through quadratic penalties. Explicit computational formulas are obtained for each stage, simplifying their software implementation. The developed method is illustrated with examples from graph theory and combinatorial optimization, including classical formulations, confirming its universality. The results of the article allow standardizing the process of adapting problems for solving on quantum annealing algorithms (e.g., D-Wave) and classical QUBO solvers.
This study proposes a novel method for simplifying inequality constraints in Higher-Order Binary Optimization (HOBO) formulations. The proposed method addresses challenges associated with Quadratic Unconstrained Binary Optimization (QUBO) formulations, specifically the increased computational complexity and reduced solution accuracy caused by the introduction of slack variables and the resulting growth in auxiliary qubits. By efficiently integrating constraints, the method enhances the computational efficiency and accuracy of both quantum and classical solvers. The effectiveness of the proposed approach is demonstrated through numerical experiments applied to combinatorial optimization problems. The results indicate that this method expands the applicability of quantum algorithms to high-dimensional problems and improves the practicality of classical optimization solvers for optimization problems involving inequality constraints.
Quadratic Unconstrained Binary Optimization models are useful for solving a diverse range of optimization problems. Constraints can be added by incorporating quadratic penalty terms into the objective, often with the introduction of slack variables needed for conversion of inequalities. This transformation can lead to a significant increase in the size and density of the problem. Herein, we propose an efficient approach for recasting inequality constraints that reduces the number of linear and quadratic variables. Experimental results illustrate the efficacy.
Quantum computing offers significant potential for solving NP-hard combinatorial (optimization) problems that are beyond the reach of classical computers. One way to tap into this potential is by reformulating combinatorial problems as a quadratic unconstrained binary optimization (QUBO) problem. The solution of the QUBO reformulation can then be addressed using adiabatic quantum computing devices or appropriate quantum computing algorithms on gate-based quantum computing devices. In general, QUBO reformulations of combinatorial problems can be readily obtained by properly penalizing the violation of the problem's constraints in the original problem's objective. However, characterizing tight (i.e., minimal but sufficient) penalty coefficients for this purpose is important and non-trivial for enabling the solution of the resulting QUBO in current and near-term quantum computing devices. Along these lines, we present closed-form characterizations of tight penalty coefficients for two distinct QUBO reformulations of the max kkk-cut problem whose values depend on the (weighted) degree of the vertices of the graph defining the problem. These findings contribute to the ongoing effort to make quantum computing a viable tool for solving combinatorial problems at scale. We support our theoretical results with illustrative examples and simple numerical results.
To solve combinatorial optimization problems more easily, it can be valuable to identify a set of necessary or sufficient conditions that an optimal solution of the problem must satisfy. For instance, weak and strong duality conditions in linear programming support the development of the well-known optimization algorithms for these problems. Similarly, the Karush–Kuhn–Tucker conditions give necessary and sufficient conditions for optimality in convex quadratic programming that underlie the development of optimization algorithms in this domain. Although such continuous conditions do not exist for integer programming, some necessary conditions can be derived from Karush–Kuhn–Tucker conditions for the Quadratic Unconstrained Binary Optimization (QUBO) problem. We present these conditions and show how they lead to a derivation of well-known criteria for fixing the values of single variables in the QUBO problem. From this, we show how to generalize these criteria to fix a product of any number of p (integer) literals, which also may be viewed as a generalization of the persistency notion, consisting of clauses of a Constraint Satisfaction Problem that the optimal solution must satisfy. We then couple our list of persistencies with state-of-the-art rules not covered by our approach. The resulting integrated set of conditions for fixing values of variables is tested in computational experiments on instances from standard databases available in the literature, showing that we can fix more variables and reduce problems more fully than previous approaches.
We extend the family of problems that may be implemented on an adiabatic quantum optimizer (AQO). When a quadratic optimization problem has at least one set of discrete controls and the constraints are linear, we call this a quadratic constrained mixed discrete optimization (QCMDO) problem. QCMDO problems are NP-hard, and no efficient classical algorithm for their solution is known. Included in the class of QCMDO problems are combinatorial optimization problems constrained by a linear partial differential equation (PDE) or system of linear PDEs. An essential complication commonly encountered in solving this type of problem is that the linear constraint may introduce many intermediate continuous variables into the optimization while the computational cost grows exponentially with problem size. We resolve this difficulty by developing a constructive mapping from QCMDO to quadratic unconstrained binary optimization (QUBO) such that the size of the QUBO problem depends only on the number of discrete control variables. With a suitable embedding, taking into account the physical constraints of the realizable coupling graph, the resulting QUBO problem can be implemented on an existing AQO. The mapping itself is efficient, scaling cubically with the number of continuous variables in the general case and linearly in the PDE case if an efficient preconditioner is available.