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Certified Algorithms: Worst-Case Analysis and Beyond

Authors Konstantin Makarychev, Yury Makarychev

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Konstantin Makarychev
  • Northwestern University, Evanston, IL, USA
Yury Makarychev
  • Toyota Technological Institute at Chicago, Chicago, IL, USA

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Konstantin Makarychev and Yury Makarychev. Certified Algorithms: Worst-Case Analysis and Beyond. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 49:1-49:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worst-case and beyond-worst-case performance guarantees. First, a γ-certified algorithm is also a γ-approximation algorithm - it finds a γ-approximation no matter what the input is. Second, it exactly solves γ-perturbation-resilient instances (γ-perturbation-resilient instances model real-life instances). Additionally, certified algorithms have a number of other desirable properties: they solve both maximization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve weakly perturbation-resilient instances, and solve optimization problems with hard constraints. In the paper, we define certified algorithms, describe their properties, present a framework for designing certified algorithms, provide examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, k-medians and k-means. We also present some negative results.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Facility location and clustering
  • certified algorithm
  • perturbation resilience
  • Bilu
  • Linial stability
  • beyond-worst-case analysis
  • approximation algorithm
  • integrality


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  1. Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. O(√log n) approximation algorithms for Min UnCut, Min 2CNF Deletion, and directed cut problems. In Proceedings of the Symposium on Theory of Computing, pages 573-581, 2005. Google Scholar
  2. Haris Angelidakis, Pranjal Awasthi, Avrim Blum, Vaggos Chatziafratis, and Chen Dan. Bilu-Linial stability, certified algorithms and the Independent Set problem. In Proceedings of the European Symposium on Algorithms, 2019. Google Scholar
  3. Haris Angelidakis, Konstantin Makarychev, and Yury Makarychev. Algorithms for stable and perturbation-resilient problems. In Proceedings of the Symposium on Theory of Computing, pages 438-451, 2017. Google Scholar
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  14. Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. SIAM Journal on Computing, 45(2):530-547, 2016. Google Scholar
  15. Konstantin Makarychev and Yury Makarychev. Bilu-Linial Stability. In T. Hazan, G. Papandreou, and D. Tarlow, editors, Perturbations, Optimization, and Statistics, chapter 13. MIT Press, 2016. Google Scholar
  16. Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Bilu-Linial stable instances of Max Cut and Minimum Multiway Cut. In Proceedings of the Symposium on Discrete Algorithms, pages 890-906, 2014. Google Scholar
  17. Ankit Sharma and Jan Vondrák. Multiway Cut, Pairwise Realizable Distributions, and Descending Thresholds. In Proceedings of the Symposium on Theory of Computing, 2014. Google Scholar
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