In reoptimization, one is given an optimal solution to a problem instance and a (locally) modified instance. The goal is to obtain a solution for the modified instance. We aim to use information obtained from the given solution in order to obtain a better solution for the new instance than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed $$\varepsilon > 0$$ ε > 0 , approximating the reoptimization problem with respect to a given $$(1+\varepsilon )$$ ( 1 + ε ) -approximation is as hard as approximating the Steiner tree problem itself. In contrast, with a given optimal solution to the original problem it is known that one can obtain considerably improved results. Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased.

中文翻译：

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Steiner 树的稳健重新优化

在重新优化中，给一个问题实例和一个（局部）修改实例的最佳解决方案。目标是为修改后的实例获得解决方案。我们的目标是使用从给定解决方案获得的信息，以便为新实例获得比我们能够从头开始计算更好的解决方案。在本文中，我们考虑 Steiner 树重新优化并解决所提供解决方案的最优性要求。我们没有假设我们得到了一个最佳解决方案，而是将假设放宽到更现实的场景，在这个场景中，我们给出了一个近似解决方案，其性能保证有上限。我们表明，对于 Steiner 树重新优化，在局部修改之间有明显的区别，其中最优性对于获得改进的近似值至关重要，而那些近似解是可接受的起点的情况。对于之前研究中考虑过的一些局部修改，我们表明对于每个固定的 $$\varepsilon > 0$$ ε > 0 ，近似于给定 $$(1+\varepsilon )$ 的重新优化问题$ ( 1 + ε ) -近似与近似 Steiner 树问题本身一样困难。相比之下，对于原始问题的给定最优解，众所周知，可以获得显着改善的结果。此外，我们提供了一种新的算法技术，有了一些进一步的见解，允许我们针对文献中考虑的所有剩余局部修改获得改进的 Steiner 树重新优化的性能保证：所需的度数大于 1 的节点成为 Steiner 节点；Steiner 节点成为必需节点；增加了一个边缘的成本。