Summary: This paper concerns lower bounding techniques for the general \(\alpha\)-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space.
This process involves two steps – (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated \(\beta\) times on an \(\alpha\)-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent \((\alpha +\beta)\)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space.
It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving \(O(N^{2(\alpha + \beta - 1)})\) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP’s. We illustrate all these concepts on the quadratic assignment problem. With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP’s of size 32 and have also improved upon existing lower bounds for other QAP’s.