Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty
| Autor: | Erlebach, Thomas, de Lima, Murilo Santos, Megow, Nicole, Schlöter, Jens |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| Popis: | We study how to utilize (possibly erroneous) predictions in a model for computing under uncertainty in which an algorithm can query unknown data. Our aim is to minimize the number of queries needed to solve the minimum spanning tree problem, a fundamental combinatorial optimization problem that has been central also to the research area of explorable uncertainty. For all integral $\gamma\ge 2$, we present algorithms that are $\gamma$-robust and $(1+\frac{1}{\gamma})$-consistent, meaning that they use at most $\gamma OPT$ queries if the predictions are arbitrarily wrong and at most $(1+\frac{1}{\gamma})OPT$ queries if the predictions are correct, where $OPT$ is the optimal number of queries for the given instance. Moreover, we show that this trade-off is best possible. Furthermore, we argue that a suitably defined hop distance is a useful measure for the amount of prediction error and design algorithms with performance guarantees that degrade smoothly with the hop distance. We also show that the predictions are PAC-learnable in our model. Our results demonstrate that untrusted predictions can circumvent the known lower bound of~$2$, without any degradation of the worst-case ratio. To obtain our results, we provide new structural insights for the minimum spanning tree problem that might be useful in the context of query-based algorithms regardless of predictions. In particular, we generalize the concept of witness sets -- the key to lower-bounding the optimum -- by proposing novel global witness set structures and completely new ways of adaptively using those. Comment: This is an extended version of an ESA 2022 paper. arXiv admin note: text overlap with arXiv:2011.07385 |
| Databáze: | arXiv |
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