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The priority model was introduced to capture "greedy-like" algorithms.Motivated by the success of advice complexity in the area of online algorithms,the fixed priority model was extended to include advice, and a reduction-basedframework was developed for proving lower bounds on the amount of advicerequired to achieve certain approximation ratios in this rather powerful model.To capture most of the algorithms that are considered greedy-like, the evenstronger model of adaptive priority algorithms is needed. We extend theadaptive priority model to include advice. We modify the reduction-basedframework from the fixed priority case to work with the more powerful adaptivepriority algorithms, simplifying the proof of correctness and strengthening allprevious lower bounds by a factor of two in the process.  We also present a purely combinatorial adaptive priority algorithm withadvice for Minimum Vertex Cover on triangle-free graphs of maximum degreethree. Our algorithm achieves optimality and uses at most 7n/22 bits of advice.No adaptive priority algorithm without advice can achieve optimality withoutadvice, and we prove that an online algorithm with advice needs more than 7n/22bits of advice to reach optimality.  We show connections between exact algorithms and priority algorithms withadvice. The branching in branch-and-reduce algorithms can be seen as trying allpossible advice strings, and all priority algorithms with advice that achieveoptimality define corresponding exact algorithms, priority exact algorithms.Lower bounds on advice-based adaptive algorithms imply lower bounds on runningtimes of exact algorithms designed in this way.

Advice Complexity of Adaptive Priority Algorithms