Introduction to special xQx issue

Abstract

The Unconstrained Binary Quadratic Programming (UBQP) problem is notable for encompassing a remarkable range of applications in combinatorial optimization. As observed in Kochenberger et al. (2013), classes of problems that can be formally expressed using UBQP formulations include: Quadratic Assignment Problems, Capital Budgeting Problems, Multiple Knapsack Problems, Task Allocation Problems (distributed computer systems), Maximum Diversity Problems, P-Median Problems, Asymmetric Assignment Problems, Symmetric Assignment Problems, Side Constrained Assignment Problems, Quadratic Knapsack Problems, Constraint Satisfaction Problems (CSPs), Set Partitioning Problems, Fixed Charge Warehouse Location Problems, Maximum Clique Problems, Maximum Independent Set Problems, Maximum Cut Problems, Graph Coloring Problems, Graph Partitioning Problems, Number Partitioning Problems, and Linear Ordering Problems. Even more remarkable is the fact that, once given a UBQP formulation, these problems can be solved by a UBQP method which is not specialized to exploit the problem domain of any individual class of problems, to yield solutions whose quality in many cases rivals or even surpasses the quality of the solutions produced by the best specialized methods, while achieving this outcome with an efficiency that likewise rivals or surpasses the efficiency of leading specialized methods. Moreover, these outcomes typically dominate those produced by current state-of-the-art commercial solvers for mixed integer linear and quadratic optimization, sometimes consuming two orders of magnitude less solution time to yield solutions superior to those of competing approaches. In other cases, where specialized methods edge out general UBQP methods for certain classes of problems, the UBQP formulation often still