Advice Complexity of Adaptive Priority Algorithms

The priority model was introduced by Borodin, Rackoff, and Nielsen (2003) to capture greedy-like algorithms. Motivated by the success of advice complexity in the area of online algorithms, Borodin et al. (2020) extended the fixed priority model to include an advice tape oracle. They also developed a reduction-based framework for proving lower bounds on the amount of advice required to achieve certain approximation ratios in this rather powerful model. In order to capture most of the algorithms that are considered greedy-like, the even stronger model of adaptive priority algorithms is needed. We extend the adaptive priority model to include an advice tape oracle. We show how to modify the reduction-based framework from the fixed priority case, making it applicable to the more powerful adaptive priority algorithms. The framework provides a template, where one can obtain a lower bound relatively easily by exhibiting "gadget patterns" fulfilling given criteria. In the process, we simplify the proof that the framework works, and we strengthen all the earlier lower bounds by a factor two. As a motivating example, we present a purely combinatorial adaptive priority algorithm with advice for Minimum Vertex Cover on triangle-free graphs of maximum degree three. Our algorithm achieves optimality and uses at most 7n/22 bits of advice. Known results imply that no adaptive priority algorithm without advice can achieve optimality without advice, and we prove that 7n/22 is fewer bits than an online algorithm with advice needs to reach optimality. Furthermore, we show connections between exact algorithms and priority algorithms with advice. Priority algorithms with advice that achieve optimality can be used to define corresponding exact algorithms, priority exact algorithms. The lower bound templates for advice-based adaptive algorithms imply lower bounds on exact algorithms designed in this way.