This research proposes a method for finding bounds in binary integer programming (BIP) and quadratic binary integer programming (QBIP). Our approach utilizes a max-flow technique within the framework of quadratic pseudo-Boolean optimization. We apply Lagrange multipliers to transform these problems into a quadratic pseudo-Boolean format, enabling us to create a corresponding network. We develop sufficient conditions under which a flow can or cannot be established in this network. The upper bound of the original problem is obtained through a feasible flow within the network. Additionally, we provide the optimality conditions for the solutions discussed in this paper. We have derived the closed forms for the various functions and values commonly used in our proposed method, as well as a lower bound for both BIP and QBIP, which can assist in solving these problems. This paper includes examples that illustrate both networks containing flows and those that do not, as well as examples that demonstrate lower and upper bound solutions as identified by our method. Furthermore, since our approach can also be applied to Quadratic Unconstrained Binary Optimization (QUBO), we provide the closed forms for the frequently used functions and values in that context, along with an example showcasing its application to a quadratic unconstrained optimization problem.
This research proposes a method for finding bounds in binary integer programming (BIP) and quadratic binary integer programming (QBIP). Our approach utilizes a max-flow technique within the framework of quadratic pseudo-Boolean optimization. We apply Lagrange multipliers to transform these problems into a quadratic pseudo-Boolean format, enabling us to create a corresponding network. We develop sufficient conditions under which a flow can or cannot be established in this network. The upper bound of the original problem is obtained through a feasible flow within the network. Additionally, we provide the optimality conditions for the solutions discussed in this paper. We have derived the closed forms for the various functions and values commonly used in our proposed method, as well as a lower bound for both BIP and QBIP, which can assist in solving these problems. This paper includes examples that illustrate both networks containing flows and those that do not, as well as examples that demonstrate lower and upper bound solutions as identified by our method. Furthermore, since our approach can also be applied to Quadratic Unconstrained Binary Optimization (QUBO), we provide the closed forms for the frequently used functions and values in that context, along with an example showcasing its application to a quadratic unconstrained optimization problem.
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.
We propose several efficient preprocessing techniques for Unconstrained Quadratic Binary Optimization (QUBO), including the direct use of enhanced versions of known basic techniques (e.g., implications derived from first and second order derivatives and from roof–duality), and the integrated use (e.g., by probing and by consensus) of the basic techniques. The application of these techniques is implemented using a natural network flow model of QUBO. The use of the proposed preprocessing techniques provides: (i) a lower bound of the minimum of the objective function, (ii) the values of some of the variables in some or every optimum, (iii) binary relations (equations, inequalities, or non-equalities) between the values of certain pairs of variables in some or every optimum, and (iv) the decomposition (if possible) of the original problem into several smaller pairwise independent QUBO problems. Extensive computational experience showing the efficiency of the proposed techniques is presented. Substantial problem simplifications, improving considerably the existing techniques, are demonstrated on benchmark problems as well as on randomly generated ones. In particular, it is shown that 100% data reduction is achieved using the proposed preprocessing techniques for MAX-CUT graphs derived from VLSI design, MAX-Clique in c-fat graphs derived from fault diagnosis, and minimum vertex cover problems in random planar graphs of up to 500 000 vertices. Acknowledgements: The third author was partially financially supported by the Portuguese FCT and by the FSE in the context of the III Quadro Comunitario de Apoio.
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.
To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer programming. State-of-the-art integer programming algorithms can compute strong relaxation bounds even for hard instances, but may have to enumerate a large number of subproblems for determining an optimum solution. If the potential of quantum computing is realized, it can be expected that in particular finding high-quality solutions for hard problems can be done fast. Still, near-future quantum hardware considerably limits the size of treatable problems. In this work, we go one step into integrating the potentials of quantum and classical techniques for combinatorial optimization. We propose a hybrid heuristic for the weighted maximum-cut problem and for quadratic unconstrained binary optimization. The heuristic employs a linear programming relaxation, rendering it well-suited for integration into exact branch-and-cut algorithms. For large instances, we reduce the problem size according to a linear relaxation such that the reduced problem can be handled by quantum machines of limited size. Moreover, we improve the applicability of depth-1 QAOA, a parameterized quantum algorithm, by deriving a parameter estimate for arbitrary instances. We present numerous computational results from real quantum hardware.
Quadratic Unconstrained Binary Optimization (QUBO) is recognized as a unifying framework for modeling a wide range of problems. Problems can be solved with commercial solvers customized for solving QUBO and since QUBO have degree two, it is useful to have a method for transforming higher degree pseudo-Boolean problems to QUBO format. The standard transformation approach requires additional auxiliary variables supported by penalty terms for each higher degree term. This paper improves on the existing cubic-to-quadratic transformation approach by minimizing the number of additional variables as well as penalty coefficient. Extensive experimental testing on Max 3-SAT modeled as QUBO shows a near 100% reduction in the subproblem size used for minimization of the number of auxiliary variables.
Quantum annealing and related combinatorial optimization methods typically accept quadratic unconstrained binary optimization (QUBO) problems as input, whereas many practical models include constraints and continuous variables. Standard QUBO conversions discretize continuous variables, increasing the binary dimension and often making feasible low-energy states harder to sample. We develop a Lagrange-multiplier method for a separable class of mixed-binary quadratic programs (MBQPs) without discretizing continuous variables. The method is based on a finite-temperature free-energy formulation and uses QUBO sampling for the binary sector. At fixed Lagrange multipliers, the continuous sector is integrated out analytically and enters only the multiplier update, leaving a QUBO over the original binary variables. We evaluate the method on the continuous relaxation of the quadratic p-median problem. Compared with a penalty-based QUBO formulation, it generates feasible solutions more reliably. At an appropriate inverse temperature, its conditional relative error is comparable to that of local search for small instances and often lower for the larger tested instances. In the time-to-target comparisons, the proposed method maintains high target-reached rates for stringent fixed-accuracy targets. For the larger instances within the tested range, it also reaches the prescribed targets faster than a commercial mixed-integer optimization solver.