Integrating combinatorial algorithms into a linear programming solver

While there are numerous linear (and nonlinear) solvers, as well as specialized algorithms for combinatorial problems, they are rarely used together. We wrote a new module for the XPRESS optimizer that lets us call the objects and functions of the LEMON C++ library. We tested this module by comparing two versions of the Dual Ascent Procedure (DAP) (Adams and Johnson in DIMACS Ser Discrete Math Theor Comput Sci 16:43–75, 1994). It is an iterative algorithm that produces lower bounds for the quadratic assignment problem (QAP). The DAP is based on the level-1 RLT-relaxation (reformulation-linearization technique) of the QAP. This formulation is significantly larger than the original QAP, but if we construct its Lagrangian dual, we can decompose the new relaxation to smaller linear assignment problems for a fixed Lagrange multiplier. The algorithm determines a better Lagrange multiplier in every iteration. We implemented two versions of the DAP with the XPRESS Optimization Software. In the first model we solved the smaller linear assignment problems with the linear programming methods of XPRESS, while in the second model we solved them by using the graph algorithms of the LEMON library via the new module. Using the instances of QAPLIB, the new module produced significantly better running times, which suggests that this direction of research might yield further results in the future.