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.

翻译：我们研究线性分配问题（LAP）对偶线性规划最优解集，并提出一种从该解集相对内部计算解的方法。假设已知任意对偶最优解和最优分配（已有多种高效算法可实现），我们的方法可在线性时间内计算出相对内点解。由于LAP作为子问题出现在二次分配问题（QAP）的线性规划松弛中，我们将该方法作为新组件引入对偶上升算法族，用于提供QAP最优值的界。为使结果适用于实际场景中具有重要应用价值的不完全QAP，我们还给出了从不完全LAP到完全LAP的线性时间归约，并附带保持最优性和相对内部性的映射。在公开基准测试上的实验表明，采用相对内点解的方案通常能提供更优的界，至少与其他方法性能相当。