（不完全）线性分配问题的相对内部解及其在二次分配问题中的应用
Relative-Interior Solution for the (Incomplete) Linear Assignment Problem with Applications to the Quadratic Assignment Problem
AI总结：
针对线性分配问题的对偶最优解集合，提出线性时间计算相对内部解的方法，并应用于二次分配问题的对偶上升算法，实验表明能提供接近LP松弛最优值的界且运行时间远低于商业求解器。
摘要：
我们研究线性分配问题（LAP）的对偶线性规划表述的最优解集合，以提出一种从该集合的相对内部计算解的方法。假设任意对偶最优解和最优分配可用（对此已存在许多高效算法），我们的方法在线性时间内计算相对内部解。由于LAP作为二次分配问题（QAP）的线性规划（LP）松弛中的子问题出现，我们将我们的方法作为对偶上升算法家族中的一个新组件，用于提供QAP最优值的界。为了使我们的结果适用于不完全QAP（在实际使用场景中备受关注），我们还提供了从不完全LAP到完全LAP的线性时间归约，以及一个保持最优性和相对内部成员资格的映射。我们在公开基准上的实验表明，使用相对内部解的方法经常能提供接近LP松弛最优值的界，并且其运行时间远低于商业LP求解器。
英文摘要：
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.
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