Relative-Interior Solution for (Incomplete) Linear Assignment Problem with Applications to Quadratic Assignment Problem
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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 LAP occurs as a subproblem in the linear programming relaxation of 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 QAP. To make our results applicable to incomplete QAP, which is of interest in practical use-cases, we also provide a linear-time reduction from incomplete LAP to 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 is frequently capable of providing superior bounds and otherwise is at least comparable.
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