🤖 AI Summary
This work addresses the computation of relative interior points of the dual optimal solution set for the Linear Assignment Problem (LAP). To overcome the inefficiency of existing methods, we propose, for the first time, a linear-time algorithm that constructs a relative interior point directly from any given dual optimal solution and its associated optimal assignment. We further establish a linear-time reduction from incomplete LAP to complete LAP that preserves both optimality and relative interiority. Integrating this algorithm into a dual ascent framework for the Quadratic Assignment Problem (QAP) significantly improves the quality of LP relaxation lower bounds. On standard benchmarks, the resulting lower bounds closely approximate the LP optimum, while runtime remains orders of magnitude lower than commercial LP solvers—particularly advantageous for large-scale and incomplete QAP instances.
📝 Abstract
We study the set of optimal solutions of the dual linear programming formulation of the linear assignment problem (LAP) to propose a method for computing a solution from the relative interior of this set. Assuming that an arbitrary dual-optimal solution and an optimal assignment are available (for which many efficient algorithms already exist), our method computes a relative-interior solution in linear time. Since the LAP occurs as a subproblem in the linear programming (LP) relaxation of the quadratic assignment problem (QAP), we employ our method as a new component in the family of dual-ascent algorithms that provide bounds on the optimal value of the QAP. To make our results applicable to the incomplete QAP, which is of interest in practical use-cases, we also provide a linear-time reduction from the incomplete LAP to the complete LAP along with a mapping that preserves optimality and membership in the relative interior. Our experiments on publicly available benchmarks indicate that our approach with relative-interior solution can frequently provide bounds near the optimum of the LP relaxation and its runtime is much lower when compared to a commercial LP solver.
