Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.30142

Strongly Regular Graph Construction via Quadratic Binary Optimization

D. Osorio, P. Andagana, V. González, J. Czartowski, D. Goyeneche

math.OCmath.COquant-phSubmitted Jul 31, 2026• Announced Aug 3, 2026• 0 citations
arXivHTMLPDF

Abstract

Strongly regular graphs, hereafter SRGs, are highly structured combinatorial objects of central interest in algebraic graph theory and quantum information, whose explicit construction becomes rapidly intractable as the number of vertices grows; Conway's still unresolved $99$-graph problem is an emblematic example. The matrix equation defining SRGs is quadratic in the binary edge variables, so the natural penalty enforcing it, a sum of squared residuals, is quartic and lies outside quadratic unconstrained binary optimization, known as QUBO. We prove a considerably stronger obstruction, namely that no quadratic function of the edge variables has the labeled SRGs as its exact set of global minimizers, except for three degenerate families given by the perfect matchings, the cocktail-party graphs and the pentagon. In particular, we formally establish that the $99$-vertex graph of Conway's problem cannot be obtained from any QUBO over the edge variables without auxiliary variables. Guided by this theorem we develop two complementary algorithms. The first operates within pure QUBO, solves the degenerate families exactly, and beyond them works as a generate--prune--verify pipeline whose output is a reduced set of candidates certified by a polynomial-time classical test. The second solves the full problem by quadratizing this quartic penalty through Rosenberg auxiliary variables, whose number grows as $Θ(N^3)$ with the number of vertices $N$. We test the pure-QUBO route under simulated annealing on all feasible families with at most $18$ vertices and the auxiliary-variable route on representative targets up to $16$ vertices, and tabulate the algebraically feasible parameter sets for graphs with at most $49$ vertices.

Comments
26 pages, 4 figures, 5 tables, 3 appendices. Includes a no-go theorem ruling out auxiliary-free QUBO encodings of strongly regular graphs, with an application to Conway's 99-graph problem. Comments welcome!

Related papers

Quadratic unconstrained binary optimization problem preprocessing: Theory and empirical analysis

Mark W. Lewis, Fred Glover

The Quadratic Unconstrained Binary Optimization problem (QUBO) has become a unifying model for representing a wide range of combinatorial optimization problems, and for linking a variety of disciplines that face these problems. A new class of quantum annealing computer that maps QUBO onto a physical qubit network structure with specific size and edge density restrictions is generating a growing interest in ways to transform the underlying QUBO structure into an equivalent graph having fewer nodes and edges. In this article, we present rules for reducing the size of the QUBO matrix by identifying variables whose value at optimality can be predetermined. We verify that the reductions improve both solution quality and time to solution and, in the case of metaheuristic methods where optimal solutions cannot be guaranteed, the quality of solutions obtained within reasonable time limits. We discuss the general QUBO structural characteristics that can take advantage of these reduction techniques and perform careful experimental design and analysis to identify and quantify the specific characteristics most affecting reduction. The rules make it possible to dramatically improve solution times on a new set of problems using both the exact Cplex solver and a tabu search metaheuristic. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(2), 79–97 2017

arXiv:1705.09844•Networks•Jun 24, 2017•104 citations

New algorithms for Quadratic Unconstrained Binary Optimization (QUBO) with applications in engineering and social sciences

Tavares, Gabriel

This dissertation investigates the Quadratic Unconstrained Binary Optimization (QUBO) problem, i.e. the problem of minimizing a quadratic function in binary variables. QUBO is studied at two complementary levels. First, there is an algorithmic aspect that tells how to preprocess the problem, how to find heuristics, how to get improved bounds and how to solve the problem with all the above ingredients.Second, there is a practical aspect that uses QUBO to solve various applications from the engineering and social sciences fields including: via minimization, 2D/3D Ising models, 1D Ising chain models, image binarization, hierarchical clustering, greedy graph coloring/partitioning, MAX-2-SAT, MIN-VC, MAX-CLIQUE, MAX-CUT, graph stability and minimum k-partition.Several families of fast heuristics for QUBO are analyzed, which include a novel probabilistic based class of methods.It is shown that there is a unique maximal set of persistencies for the linearization model for QUBO.This set is determined in polynomial time by a maximum flow followed by the computation of the strong components of a network that has 2n+2 nodes, where n is the number of variables. The identification of the above persistencies leads to a unique decomposition of the function, such that each component can be optimized separately. To find further persistencies, two additional techniques are proposed: one is based on the second order derivatives of Hammer et al. [HH81]; the other technique is a probing procedure on the two possible values of the variables.These preprocessing tools work remarkably well for certain classes of problems.We improved the Iterated Roof-Dual bound (IRD) of [BH89] by proposing two combinatorial methods: one was named the squeezed IRD; and the second was called the project-and-lift IRD method.The cubic-dual bound can be found by means of linear programming by adding a set of triangle inequalities to the standard linearization, whose number is cubic in the number of variables. We show that this set can be reduced depending on the coefficients of the terms of the function. This leads to the possibility of computing the cubic-duals of larger QUBOs.

•Jan 1, 2008•7 citations

Logical and Inequality Implications for Reducing the Size and Complexity of Quadratic Unconstrained Binary Optimization Problems

Fred Glover, Mark W. Lewis, Gary A. Kochenberger

The quadratic unconstrained binary optimization (QUBO) problem arises in diverse optimization applications ranging from Ising spin problems to classical problems in graph theory and binary discrete optimization. The use of preprocessing to transform the graph representing the QUBO problem into a smaller equivalent graph is important for improving solution quality and time for both exact and metaheuristic algorithms and is a step towards mapping large scale QUBO to hardware graphs used in quantum annealing computers. In an earlier paper (Lewis and Glover, 2016) a set of rules was introduced that achieved significant QUBO reductions as verified through computational testing. Here this work is extended with additional rules that provide further reductions that succeed in exactly solving 10% of the benchmark QUBO problems. An algorithm and associated data structures to efficiently implement the entire set of rules is detailed and computational experiments are reported that demonstrate their efficacy.

arXiv:1705.09545•May 26, 2017•0 citations

A penalty-free QUBO formulation for the minimum maximal matching problem

Yozo Nakahara, Shunsuke Tsukiyama, Koji Nakano +2 more

Given an undirected graph, a matching is a set of edges with no shared vertices. A maximal matching cannot be extended by adding edges without violating this condition. The minimum maximal matching problem seeks the smallest such matching. While it can be formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem, existing methods require complex penalty terms. This paper presents a novel, penalty-free QUBO formulation that is simpler and more intuitive. We analyze its theoretical properties and evaluate its performance using several QUBO solvers. Our approach enhances the practicality of QUBO-based optimization for combinatorial graph problems.

International Journal of Parallel Emergent and Distributed Systems•Nov 7, 2025•0 citations

Solving Quadratic Unconstrained Binary Optimization with divide-and-conquer and quantum algorithms

Gian Giacomo Guerreschi

Quadratic Unconstrained Binary Optimization (QUBO) is a broad class of optimization problems with many practical applications. To solve its hard instances in an exact way, known classical algorithms require exponential time and several approximate methods have been devised to reduce such cost. With the growing maturity of quantum computing, quantum algorithms have been proposed to speed up the solution by using either quantum annealers or universal quantum computers. Here we apply the divide-and-conquer approach to reduce the original problem to a collection of smaller problems whose solutions can be assembled to form a single Polynomial Binary Unconstrained Optimization instance with fewer variables. This technique can be applied to any QUBO instance and leads to either an all-classical or a hybrid quantum-classical approach. When quantum heuristics like the Quantum Approximate Optimization Algorithm (QAOA) are used, our proposal leads to a double advantage: a substantial reduction of quantum resources, specifically an average of ~42% fewer qubits to solve MaxCut on random 3-regular graphs, together with an improvement in the quality of the approximate solutions reached.

arXiv:2101.07813•Jan 19, 2021•12 citations

REDUCING DISCRETE OPTIMIZATION PROBLEMS TO THE QUBO FORM

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

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.

Problems of Information Transmission•Jan 1, 2025•2 citations
Strongly regular graphs, hereafter SRGs, are highly structured combinatorial objects of central interest in algebraic graph theory and quantum information, whose explicit construction becomes rapidly intractable as the number of vertices grows; Conway's still unresolved 999999-graph problem is an emblematic example. The matrix equation defining SRGs is quadratic in the binary edge variables, so the natural penalty enforcing it, a sum of squared residuals, is quartic and lies outside quadratic unconstrained binary optimization, known as QUBO. We prove a considerably stronger obstruction, namely that no quadratic function of the edge variables has the labeled SRGs as its exact set of global minimizers, except for three degenerate families given by the perfect matchings, the cocktail-party graphs and the pentagon. In particular, we formally establish that the 999999-vertex graph of Conway's problem cannot be obtained from any QUBO over the edge variables without auxiliary variables. Guided by this theorem we develop two complementary algorithms. The first operates within pure QUBO, solves the degenerate families exactly, and beyond them works as a generate--prune--verify pipeline whose output is a reduced set of candidates certified by a polynomial-time classical test. The second solves the full problem by quadratizing this quartic penalty through Rosenberg auxiliary variables, whose number grows as Θ(N3)Θ(N^3)Θ(N3) with the number of vertices NNN. We test the pure-QUBO route under simulated annealing on all feasible families with at most 181818 vertices and the auxiliary-variable route on representative targets up to 161616 vertices, and tabulate the algebraically feasible parameter sets for graphs with at most 494949 vertices.
The Quadratic Unconstrained Binary Optimization problem (QUBO) has become a unifying model for representing a wide range of combinatorial optimization problems, and for linking a variety of disciplines that face these problems. A new class of quantum annealing computer that maps QUBO onto a physical qubit network structure with specific size and edge density restrictions is generating a growing interest in ways to transform the underlying QUBO structure into an equivalent graph having fewer nodes and edges. In this article, we present rules for reducing the size of the QUBO matrix by identifying variables whose value at optimality can be predetermined. We verify that the reductions improve both solution quality and time to solution and, in the case of metaheuristic methods where optimal solutions cannot be guaranteed, the quality of solutions obtained within reasonable time limits. We discuss the general QUBO structural characteristics that can take advantage of these reduction techniques and perform careful experimental design and analysis to identify and quantify the specific characteristics most affecting reduction. The rules make it possible to dramatically improve solution times on a new set of problems using both the exact Cplex solver and a tabu search metaheuristic. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(2), 79–97 2017
This dissertation investigates the Quadratic Unconstrained Binary Optimization (QUBO) problem, i.e. the problem of minimizing a quadratic function in binary variables. QUBO is studied at two complementary levels. First, there is an algorithmic aspect that tells how to preprocess the problem, how to find heuristics, how to get improved bounds and how to solve the problem with all the above ingredients.Second, there is a practical aspect that uses QUBO to solve various applications from the engineering and social sciences fields including: via minimization, 2D/3D Ising models, 1D Ising chain models, image binarization, hierarchical clustering, greedy graph coloring/partitioning, MAX-2-SAT, MIN-VC, MAX-CLIQUE, MAX-CUT, graph stability and minimum k-partition.Several families of fast heuristics for QUBO are analyzed, which include a novel probabilistic based class of methods.It is shown that there is a unique maximal set of persistencies for the linearization model for QUBO.This set is determined in polynomial time by a maximum flow followed by the computation of the strong components of a network that has 2n+2 nodes, where n is the number of variables. The identification of the above persistencies leads to a unique decomposition of the function, such that each component can be optimized separately. To find further persistencies, two additional techniques are proposed: one is based on the second order derivatives of Hammer et al. [HH81]; the other technique is a probing procedure on the two possible values of the variables.These preprocessing tools work remarkably well for certain classes of problems.We improved the Iterated Roof-Dual bound (IRD) of [BH89] by proposing two combinatorial methods: one was named the squeezed IRD; and the second was called the project-and-lift IRD method.The cubic-dual bound can be found by means of linear programming by adding a set of triangle inequalities to the standard linearization, whose number is cubic in the number of variables. We show that this set can be reduced depending on the coefficients of the terms of the function. This leads to the possibility of computing the cubic-duals of larger QUBOs.
The quadratic unconstrained binary optimization (QUBO) problem arises in diverse optimization applications ranging from Ising spin problems to classical problems in graph theory and binary discrete optimization. The use of preprocessing to transform the graph representing the QUBO problem into a smaller equivalent graph is important for improving solution quality and time for both exact and metaheuristic algorithms and is a step towards mapping large scale QUBO to hardware graphs used in quantum annealing computers. In an earlier paper (Lewis and Glover, 2016) a set of rules was introduced that achieved significant QUBO reductions as verified through computational testing. Here this work is extended with additional rules that provide further reductions that succeed in exactly solving 10% of the benchmark QUBO problems. An algorithm and associated data structures to efficiently implement the entire set of rules is detailed and computational experiments are reported that demonstrate their efficacy.
Given an undirected graph, a matching is a set of edges with no shared vertices. A maximal matching cannot be extended by adding edges without violating this condition. The minimum maximal matching problem seeks the smallest such matching. While it can be formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem, existing methods require complex penalty terms. This paper presents a novel, penalty-free QUBO formulation that is simpler and more intuitive. We analyze its theoretical properties and evaluate its performance using several QUBO solvers. Our approach enhances the practicality of QUBO-based optimization for combinatorial graph problems.
Quadratic Unconstrained Binary Optimization (QUBO) is a broad class of optimization problems with many practical applications. To solve its hard instances in an exact way, known classical algorithms require exponential time and several approximate methods have been devised to reduce such cost. With the growing maturity of quantum computing, quantum algorithms have been proposed to speed up the solution by using either quantum annealers or universal quantum computers. Here we apply the divide-and-conquer approach to reduce the original problem to a collection of smaller problems whose solutions can be assembled to form a single Polynomial Binary Unconstrained Optimization instance with fewer variables. This technique can be applied to any QUBO instance and leads to either an all-classical or a hybrid quantum-classical approach. When quantum heuristics like the Quantum Approximate Optimization Algorithm (QAOA) are used, our proposal leads to a double advantage: a substantial reduction of quantum resources, specifically an average of ~42% fewer qubits to solve MaxCut on random 3-regular graphs, together with an improvement in the quality of the approximate solutions reached.
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.