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Advice Complexity of Priority Algorithms

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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.

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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. Consider \(D=\mathbb {R}\times \mathbb {R}\) with the lexicographic ordering. Assume to the contrary that f is an order-embedding mapping from D to \(\mathbb {R}\). Then each subset of D of the form \(r \times \mathbb {R}\), where \(r\in \mathbb {R}\), has to be mapped into an interval of \(\mathbb {R}\) which is disjoint from any other subset \(r^{\prime }\times \mathbb {R}\) for \(r\not = r^{\prime }\). Thus, f defines an uncountable number of disjoint intervals of \(\mathbb {R}\). At the same time, between any two reals, we can find a rational number (by considering the position where the two reals differ for the first time). Using this on the end points of the intervals above, each interval must contain a rational number which does not appear in any other interval. Thus, there are an uncountable number of rational numbers, which is a contradiction. (This argument appears in [22].)

  3. In Theorem 3 and in all of our lower bound advice results, we state the result so as to include \(\epsilon = \frac {1}{2}\), in which case the conditions “fewer than (1/2 − 𝜖)” and “fewer than (1 − H(𝜖))” make the statements vacuously true.

  4. 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.

  5. However, both gadgets within a pair do not necessarily have the same topological structure. In Triangle Finding, they did not.

References

  1. Alekhnovich, M., Borodin, A., Buresh-Oppenheim, J., Impagliazzo, R., Magen, A., Pitassi, T.: Toward a model for backtracking and dynamic programming. Comput. Complex. 20(4), 679–740 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angelopoulos, S., Borodin, A.: On the power of priority algorithms for facility location and set cover. Algorithmica 40(4), 271–291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Besser, B., Poloczek, M.: Greedy matching: guarantees and limitations. Algorithmica 77(1), 201–234 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Böckenhauer, H.J., Hromkovič, J., Komm, D., Krug, S., Smula, J., Sprock, A.: The string guessing problem as a method to prove lower bounds on the advice complexity. Theoretical Comput. Sci. 554, 95–108 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Böckenhauer, H.J., Komm, D., Královič, R., Královič, R.: On the advice complexity of the K-server problem. In: 38th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 6755, pp 207–218. Springer (2011)

  6. Böckenhauer, H.J., Komm, D., Královič, R., Královič, R., Mömke, T.: Online algorithms with advice: the tape model. Inf. Comput. 254(1), 59–83 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Borodin, A., Boyar, J., Larsen, K.S., Mirmohammadi, N.: Priority algorithms for graph optimization problems. Theor. Comput. Sci. 411(1), 239–258 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Borodin, A., Boyar, J., Larsen, K.S., Pankratov, D.: Advice complexity of priority algorithms. In: 16th International Workshop on Approximation and Online Algorithms (WAOA), Lecture Notes in Computer Science, vol. 11312, pp 69–86. Springer (2018)

  9. Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  10. Borodin, A., Lucier, B.: On the limitations of greedy mechanism design for truthful combinatorial auctions. ACM Trans. Econ. Comput. 5(1), 2:1–2:23 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Borodin, A., Nielsen, M.N., Rackoff, C.: (Incremental) priority algorithms. Algorithmica 37(4), 295–326 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Boyar, J., Favrholdt, L.M., Kudahl, C., Larsen, K.S., Mikkelsen, J.W.: Online algorithms with advice: a survey. ACM Comput. Surveys 50(2), 19:1–19:34 (2017)

    Article  MATH  Google Scholar 

  13. Boyar, J., Kamali, S., Larsen, K.S., López-ortiz, A.: Online bin packing with advice. Algorithmica 74(1), 507–527 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cook, S.A.: The complexity of theorem-proving procedures. In: 3rd Annual ACM Symposium on Theory of Computing (STOC), pp 151–158. ACM (1971)

  15. Davis, S., Impagliazzo, R.: Models of greedy algorithms for graph problems. Algorithmica 54(3), 269–317 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dürr, C., Konrad, C., Renault, M.P.: On the power of advice and randomization for online bipartite matching. In: 24th Annual European Symposium on Algorithms (ESA), LIPIcs, vol. 57, pp 37:1–37:16. Schloss Dagstuhl–Leibniz-Zentrum Fuer Informatik (2016)

  17. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, the IBM Research Symposia Series, pp 85–103 (1972)

  18. Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: 22nd Annual ACM Symposium on Theory of Computing (STOC), pp 352–358. ACM (1990)

  19. Komm, D.: An Introduction to Online Computation – Determinism, Randomization, Advice. Texts in Theoretical Computer Science. An EATCS Series. Springer (2016)

  20. Lesh, N., Mitzenmacher, M.: Bubblesearch: a simple heuristic for improving priority-based greedy algorithms. Inf. Process. Lett. 97(4), 161–169 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lund, C., Yannakakis, M.: The approximation of maximum subgraph problems. In: 20th International Colloquium on Automata, Languages and Programming (ICALP), LNCS, vol. 700, pp 40–51. Springer, Berlin (1993)

  22. Mathematics StackExchange: Is the set of real numbers the largest possible totally ordered set? https://math.stackexchange.com/questions/255145/is-the-set-of-real-numbers-the-largest-possible-totally-ordered-set https://math.stackexchange.com/questions/255145/is-the-set-of-real-numbers-the-largest-possible-totally-ordered-set. Accessed 31 July 2019 (2019)

  23. McGregor, A.: Graph stream algorithms: a survey. ACM SIGMOD Record 43 (1), 9–20 (2014)

    Article  Google Scholar 

  24. Mikkelsen, J.W.: Randomization can be as helpful as a glimpse of the future in online computation. In: 43rd International Colloquium on Automata, Languages and Programming (ICALP), LIPIcs. The details can be found in arXiv:1511.05886[cs.DS], vol. 55, pp 39:1–39:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)

  25. Pena, N., Borodin, A.: On the limitations of deterministic de-randomizations for online bipartite matching and max-sat. arXiv:1608.03182[cs.DS] (2016)

  26. Pena, N., Borodin, A.: On extensions of the deterministic online model for bipartite matching and max-sat. Theor. Comput. Sci. 770, 1–24 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Poloczek, M.: Bounds on greedy algorithms for MAX SAT. In: 19th Annual European Symposium on Algorithms (ESA), LNCS, vol. 6942, pp 37–48. Springer (2011)

  28. Poloczek, M., Schnitger, G., Williamson, D.P., van Zuylen, A.: Greedy algorithms for the maximum satisfiability problem: Simple algorithms and inapproximability bounds. SIAM J Comput 46(3), 1029–1061 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Regev, O.: Priority algorithms for makespan minimization in the subset model. Inf. Process. Lett. 84(3), 153–157 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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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Authors and Affiliations

  1. Department of Computer Science, University of Toronto, Toronto, Canada

    Allan Borodin

  2. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

    Joan Boyar & Kim S. Larsen

  3. Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada

    Denis Pankratov

Authors
  1. Allan Borodin
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  2. Joan Boyar
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  3. Kim S. Larsen
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  4. Denis Pankratov
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Correspondence to Kim S. Larsen.

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This article belongs to the Topical Collection: Special Issue on Approximation and Online Algorithms 2018

Guest Editors: Leah Epstein and Thomas Erlebach

The first author was supported by the Natural Sciences and Engineering Research Council of Canada. The second and third authors were supported in part by the Independent Research Fund Denmark, Natural Sciences, grant DFF-7014-00041. A preliminary version of this paper was presented at WAOA 2018 [8]. This journal version is extended significantly more than 30% and contains more details in general, proofs of all theorems, and new sections on Maximum Cut, Maximum Satisfiability, Job Scheduling, and Vertex Cover.

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Borodin, A., Boyar, J., Larsen, K.S. et al. Advice Complexity of Priority Algorithms. Theory Comput Syst 64, 593–625 (2020). https://doi.org/10.1007/s00224-019-09955-7

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