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Approximation Algorithms for 𝓁_p-Shortest Path and 𝓁_p-Group Steiner Tree

Authors Yury Makarychev , Max Ovsiankin , Erasmo Tani

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Shortest paths
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Routing and network design problems
Keywords
  • Shortest Path
  • Asymmetric Group Steiner Tree
  • Sum-of-Squares

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Abstract

We present polylogarithmic approximation algorithms for variants of the Shortest Path, Group Steiner Tree, and Group ATSP problems with vector costs. In these problems, each edge e has a vector cost c_e ∈ ℝ_{≥0}^𝓁. For a feasible solution - a path, subtree, or tour (respectively) - we find the total vector cost of all the edges in the solution and then compute the 𝓁_p-norm of the obtained cost vector (we assume that p ≥ 1 is an integer). Our algorithms for series-parallel graphs run in polynomial time and those for arbitrary graphs run in quasi-polynomial time.
To obtain our results, we introduce and use new flow-based Sum-of-Squares relaxations. We also obtain a number of hardness results.

Cite As Get BibTex

Yury Makarychev, Max Ovsiankin, and Erasmo Tani. Approximation Algorithms for 𝓁_p-Shortest Path and 𝓁_p-Group Steiner Tree. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 111:1-111:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.111

Author Details

Yury Makarychev
  • Toyota Technological Institute at Chicago, IL, USA
Max Ovsiankin
  • Toyota Technological Institute at Chicago, IL, USA
Erasmo Tani
  • University of Chicago, IL, USA

Funding

  • Makarychev, Yury: Partially supported by NSF Awards CCF-1955173, CCF-1934843, and ECCS-2216899.
  • Ovsiankin, Max: Partially supported by the Institute for Data, Econometrics, Algorithms, and Learning (IDEAL) with NSF Grant ECCS-2216899.
  • Tani, Erasmo: Partially supported by the Institute for Data, Econometrics, Algorithms, and Learning (IDEAL) with NSF Grant ECCS-2216912.

Acknowledgements

We would like to thank the anonymous reviewers for providing references to relevant prior work and for comments that improved the presentation of this paper.

References

  1. Hassene Aissi, Cristina Bazgan, and Daniel Vanderpooten. Approximation of min-max and min-max regret versions of some combinatorial optimization problems. European Journal of Operational Research, 179(2):281-290, 2007. URL: https://doi.org/10.1016/j.ejor.2006.03.023.
  2. Boaz Barak, Jonathan A. Kelner, and David Steurer. Rounding sum-of-squares relaxations. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 31-40. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591886.
  3. Vittorio Bilò, Ioannis Caragiannis, Angelo Fanelli, Michele Flammini, and Gianpiero Monaco. Simple greedy algorithms for fundamental multidimensional graph problems. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 125:1-125:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.125.
  4. Thomas Breugem, Twan Dollevoet, and Wilco van den Heuvel. Analysis of fptases for the multi-objective shortest path problem. Comput. Oper. Res., 78:44-58, 2017. URL: https://doi.org/10.1016/j.cor.2016.06.022.
  5. Moses Charikar, Chandra Chekuri, To-Yat Cheung, Zuo Dai, Ashish Goel, Sudipto Guha, and Ming Li. Approximation algorithms for directed steiner problems. Journal of Algorithms, 33(1):73-91, 1999. URL: https://doi.org/10.1006/jagm.1999.1042.
  6. Moses Charikar, Chandra Chekuri, Ashish Goel, and Sudipto Guha. Rounding via trees: deterministic approximation algorithms for group steiner trees and k-median. In Proceedings of the Symposium on Theory of Computing, pages 114-123, 1998. URL: https://doi.org/10.1145/276698.276719.
  7. Chandra Chekuri and Martin Pál. A recursive greedy algorithm for walks in directed graphs. In Proceedings of the Symposium on Foundations of Computer Science, pages 245-253, 2005. URL: https://doi.org/10.1109/SFCS.2005.9.
  8. Chandra Chekuri, Jan Vondrák, and Rico Zenklusen. Dependent randomized rounding via exchange properties of combinatorial structures. In Proceedings of the Symposium on Foundations of Computer Science, pages 575-584, 2010. URL: https://doi.org/10.1109/FOCS.2010.60.
  9. Julia Chuzhoy and Sanjeev Khanna. Hardness of directed routing with congestion. Electron. Colloquium Comput. Complex., TR06-109(109), 2006. URL: https://arxiv.org/abs/TR06-109.
  10. Jesús de la Cal and Javier Cárcamo. Set partitions and moments of random variables. Journal of mathematical analysis and applications, 378(1):16-22, 2011. Google Scholar
  11. Matthias Ehrgott. Multicriteria Optimization (2. ed.), volume 491. Springer, 2005. URL: https://doi.org/10.1007/3-540-27659-9.
  12. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. In Proceedings of the Symposium on Theory of Computing, pages 448-455, 2003. URL: https://doi.org/10.1145/780542.780608.
  13. Noah Fleming, Pravesh Kothari, and Toniann Pitassi. Semialgebraic proofs and efficient algorithm design. Foundations and Trends in Theoretical Computer Science, 14(1-2):1-221, 2019. URL: https://doi.org/10.1561/0400000086.
  14. Naveen Garg, Goran Konjevod, and Ramamoorthi Ravi. A polylogarithmic approximation algorithm for the group steiner tree problem. Journal of Algorithms, 37(1):66-84, 2000. URL: https://doi.org/10.1006/jagm.2000.1096.
  15. Horst W Hamacher and Günter Ruhe. On spanning tree problems with multiple objectives. Annals of Operations Research, 52(4):209-230, 1994. URL: https://doi.org/10.1007/BF02032304.
  16. Pierre Hansen. Bicriterion path problems. In Multiple Criteria Decision Making Theory and Application: Proceedings of the Third Conference Hagen/Königswinter, West Germany, August 20-24, 1979, pages 109-127. Springer, 1980. Google Scholar
  17. Jovan Karamata. Sur une inégalité relative aux fonctions convexes. Publications de l'Institut mathematique, 1(1):145-147, 1932. Google Scholar
  18. Adam Kasperski and Paweł Zieliński. On the approximability of minmax (regret) network optimization problems. Information Processing Letters, 109(5):262-266, 2009. URL: https://doi.org/10.1016/j.ipl.2008.10.008.
  19. Adam Kasperski and Paweł Zieliński. On the approximability of robust spanning tree problems. Theoretical Computer Science, 412(4-5):365-374, 2011. URL: https://doi.org/10.1016/j.tcs.2010.10.006.
  20. Adam Kasperski and Paweł Zieliński. Robust discrete optimization under discrete and interval uncertainty: A survey. Robustness analysis in decision aiding, optimization, and analytics, pages 113-143, 2016. Google Scholar
  21. Adam Kasperski and Pawel Zielinski. Approximating some network problems with scenarios. CoRR, abs/1806.08936, 2018. URL: https://doi.org/10.48550/arXiv.1806.08936.
  22. Aditi Laddha, Mohit Singh, and Santosh S Vempala. Socially fair network design via iterative rounding. Operations Research Letters, 50(5):536-540, 2022. URL: https://doi.org/10.1016/j.orl.2022.07.011.
  23. Shi Li, Chenyang Xu, and Ruilong Zhang. Polylogarithmic approximation for robust s-t path. CoRR, abs/2305.16439, 2023. URL: https://doi.org/10.48550/arXiv.2305.16439.
  24. Yury Makarychev, Max Ovsiankin, and Erasmo Tani. Approximation algorithms for 𝓁_p-shortest path and 𝓁_p-group steiner tree, 2024. URL: https://arxiv.org/abs/2404.17669.
  25. Ernesto Queiros Vieira Martins. On a multicriteria shortest path problem. European Journal of Operational Research, 16(2):236-245, 1984. Google Scholar
  26. Robert Franklin Muirhead. Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proceedings of the Edinburgh Mathematical Society, 21:144-162, 1902. Google Scholar
  27. Prasad Raghavendra and Benjamin Weitz. On the bit complexity of sum-of-squares proofs. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 80:1-80:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.80.
  28. R. Ravi, Madhav V. Marathe, S. S. Ravi, Daniel J. Rosenkrantz, and Harry B. Hunt III. Many birds with one stone: multi-objective approximation algorithms. In S. Rao Kosaraju, David S. Johnson, and Alok Aggarwal, editors, Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, May 16-18, 1993, San Diego, CA, USA, pages 438-447. ACM, 1993. URL: https://doi.org/10.1145/167088.167209.
  29. Gabriele Reich and Peter Widmayer. Beyond steiner’s problem: A vlsi oriented generalization. In International Workshop on Graph-theoretic Concepts in Computer Science, pages 196-210. Springer, 1989. URL: https://doi.org/10.1007/3-540-52292-1_14.
  30. Stefan Ruzika and Horst W Hamacher. A survey on multiple objective minimum spanning tree problems. In Algorithmics of Large and Complex Networks: Design, Analysis, and Simulation, pages 104-116. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02094-0_6.
  31. Walter J Savitch. Relationships between nondeterministic and deterministic tape complexities. Journal of computer and system sciences, 4(2):177-192, 1970. URL: https://doi.org/10.1016/S0022-0000(70)80006-X.
  32. Richard Stanley. Enumerative Combinatorics: Volume 2. Cambridge University Press, 2023. Google Scholar
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