New algorithms for quadratic unconstrained binary optimization (qubo) with applications in engineering and social sciences

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. [121]; 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 [51] 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.