Abstract
The priority model of “greedy-like” algorithms was introduced by Borodin, Nielsen, and Rackoff in 2002. We augment this model by allowing priority algorithms to have access to advice, i.e., side information precomputed by an all-powerful oracle. Obtaining lower bounds in the priority model without advice can be challenging and may involve intricate adversary arguments. Since the priority model with advice is even more powerful, obtaining lower bounds presents additional difficulties. We sidestep these difficulties by developing a general framework of reductions which makes lower-bound proofs relatively straightforward and routine. We start by introducing the Pair Matching problem, for which we are able to prove strong lower bounds in the priority model with advice. We develop a template for constructing a reduction from Pair Matching to other problems in the priority model with advice – this part is technically challenging since the reduction needs to define a valid priority function for Pair Matching while respecting the priority function for the other problem. Finally, we apply the template to obtain lower bounds for a number of standard discrete optimization problems.
The full version of the paper is available on arXiv [6]. For the first author, research is supported by NSERC. The second and third authors were supported in part by the Independent Research Fund Denmark, Natural Sciences, grant DFF-7014-00041.
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Notes
- 1.
In the adaptive priority model, the algorithm is allowed to specify a new ordering depending on previous items and decisions before a new input item is presented.
- 2.
In Theorem 3 and in all of our lower bound advice results, we state the result so as to include \(\varepsilon = \frac{1}{2}\), in which case the conditions “fewer than \((1/2-\varepsilon )\)” and “fewer than \((1-H(\varepsilon ))\)” make the statements vacuously true.
- 3.
There are similarities to the NP-Complete problems, Numerical Matching with Target Sums and Numerical 3-Dimensional Matching, though these problems ask if permutations of sets of inputs will lead to a complete matching.
- 4.
However, both gadgets within a pair do not necessarily have the same topological structure. In Triangle Finding, they did not.
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Acknowledgments
Part of the work was done when the first author was visiting Toyota Technological Institute at Chicago. The work was initiated while the second and third authors were visiting the University of Toronto. Most of the work was done when the fourth author was a postdoc at the University of Toronto.
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Borodin, A., Boyar, J., Larsen, K.S., Pankratov, D. (2018). Advice Complexity of Priority Algorithms. In: Epstein, L., Erlebach, T. (eds) Approximation and Online Algorithms. WAOA 2018. Lecture Notes in Computer Science(), vol 11312. Springer, Cham. https://doi.org/10.1007/978-3-030-04693-4_5
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