Polylogarithmic Approximations for Robust s-t Path

Authors Shi Li , Chenyang Xu , Ruilong Zhang

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Shi Li
  • Department of Computer Science and Technology, Nanjing University, Jiangsu, China
Chenyang Xu
  • Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, China
Ruilong Zhang
  • Department of Computer Science, City University of Hong Kong, Hong Kong, China

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Shi Li, Chenyang Xu, and Ruilong Zhang. Polylogarithmic Approximations for Robust s-t Path. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 106:1-106:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


The paper revisits the Robust s-t Path problem, one of the most fundamental problems in robust optimization. In the problem, we are given a directed graph with n vertices and k distinct cost functions (scenarios) defined over edges, and aim to choose an s-t path such that the total cost of the path is always provable no matter which scenario is realized. Viewing each cost function as an agent, our goal is to find a fair s-t path, which minimizes the maximum cost among all agents. The problem is NP-hard to approximate within a factor of o(log k) unless NP ⊆ DTIME(n^{polylog n}), and the best-known approximation ratio is Õ(√n), which is based on the natural flow linear program. A longstanding open question is whether we can achieve a polylogarithmic approximation for the problem; it remains open even if a quasi-polynomial running time is allowed. Our main result is a O(log n log k) approximation for the Robust s-t Path problem in quasi-polynomial time, solving the open question in the quasi-polynomial time regime. The algorithm is built on a novel linear program formulation for a decision-tree-type structure, which enables us to overcome the Ω(√n) integrality gap for the natural flow LP. Furthermore, we show that for graphs with bounded treewidth, the quasi-polynomial running time can be improved to a polynomial. We hope our techniques can offer new insights into this problem and other related problems in robust optimization.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Approximation Algorithm
  • Randomized LP Rounding
  • Robust s-t Path


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  1. Jacob D. Abernethy, Pranjal Awasthi, Matthaus Kleindessner, Jamie Morgenstern, Chris Russell, and Jie Zhang. Active sampling for min-max fairness. In ICML, volume 162 of Proceedings of Machine Learning Research, pages 53-65. PMLR, 2022. Google Scholar
  2. Hassene Aissi, Cristina Bazgan, and Daniel Vanderpooten. Approximation of min-max and min-max regret versions of some combinatorial optimization problems. Eur. J. Oper. Res., 179(2):281-290, 2007. Google Scholar
  3. Hassene Aissi, Cristina Bazgan, and Daniel Vanderpooten. Min-max and min-max regret versions of combinatorial optimization problems: A survey. European journal of operational research, 197(2):427-438, 2009. Google Scholar
  4. Hassene Aissi, Cristina Bazgan, and Daniel Vanderpooten. General approximation schemes for min-max (regret) versions of some (pseudo-)polynomial problems. Discret. Optim., 7(3):136-148, 2010. Google Scholar
  5. Yang An and Rui Gao. Generalization bounds for (wasserstein) robust optimization. In NeurIPS, pages 10382-10392, 2021. Google Scholar
  6. Aharon Ben-Tal, Laurent El Ghaoui, and Arkadi Nemirovski. Robust Optimization, volume 28 of Princeton Series in Applied Mathematics. Princeton University Press, 2009. Google Scholar
  7. Vittorio Bilò, Ioannis Caragiannis, Angelo Fanelli, Michele Flammini, and Gianpiero Monaco. Simple greedy algorithms for fundamental multidimensional graph problems. In ICALP, volume 80 of LIPIcs, pages 125:1-125:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  8. Hans L. Bodlaender. Nc-algorithms for graphs with small treewidth. In WG, volume 344 of Lecture Notes in Computer Science, pages 1-10. Springer, 1988. Google Scholar
  9. Chandra Chekuri, Jan Vondrák, and Rico Zenklusen. Dependent randomized rounding via exchange properties of combinatorial structures. In FOCS, pages 575-584. IEEE Computer Society, 2010. Google Scholar
  10. Qingyun Chen, Sungjin Im, Benjamin Moseley, Chenyang Xu, and Ruilong Zhang. Min-max submodular ranking for multiple agents. AAAI 2023, to appear, 2023. Google Scholar
  11. Altannar Chinchuluun and Panos M. Pardalos. A survey of recent developments in multiobjective optimization. Ann. Oper. Res., 154(1):29-50, 2007. Google Scholar
  12. Vincent Conitzer, Rupert Freeman, and Nisarg Shah. Fair public decision making. In EC, pages 629-646. ACM, 2017. Google Scholar
  13. Michael Dinitz, Guy Kortsarz, and Shi Li. Degrees and network design: New problems and approximations. CoRR, abs/2302.11475, 2023. URL: https://arxiv.org/abs/2302.11475.
  14. Daniel Duque, Leonardo Lozano, and Andrés L. Medaglia. An exact method for the biobjective shortest path problem for large-scale road networks. Eur. J. Oper. Res., 242(3):788-797, 2015. Google Scholar
  15. Matthias Ehrgott. Multicriteria Optimization (2. ed.). Springer, 2005. Google Scholar
  16. Brandon Fain, Kamesh Munagala, and Nisarg Shah. Fair allocation of indivisible public goods. In EC, pages 575-592. ACM, 2018. Google Scholar
  17. Virginie Gabrel, Cécile Murat, and Aurélie Thiele. Recent advances in robust optimization: An overview. Eur. J. Oper. Res., 235(3):471-483, 2014. Google Scholar
  18. Yu Gang and Yang Jian. On the robust shortest path problem. Comput. Oper. Res., 25(6):457-468, 1998. Google Scholar
  19. Fabrizio Grandoni, Bundit Laekhanukit, and Shi Li. O(log^2 k / log log k)-approximation algorithm for directed steiner tree: a tight quasi-polynomial-time algorithm. In STOC, pages 253-264. ACM, 2019. Google Scholar
  20. Adam Kasperski and Pawel Zielinski. On the approximability of minmax (regret) network optimization problems. Inf. Process. Lett., 109(5):262-266, 2009. Google Scholar
  21. Adam Kasperski and Pawel Zielinski. On the approximability of robust spanning tree problems. Theor. Comput. Sci., 412(4-5):365-374, 2011. Google Scholar
  22. 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
  23. Adam Kasperski and Pawel Zielinski. Approximating some network problems with scenarios. CoRR, abs/1806.08936, 2018. URL: https://arxiv.org/abs/1806.08936.
  24. Ana Klobucar and Robert Manger. Solving robust weighted independent set problems on trees and under interval uncertainty. Symmetry, 13(12):2259, 2021. Google Scholar
  25. Shi Li, Chenyang Xu, and Ruilong Zhang. Polylogarithmic approximation for robust s-t path. CoRR, abs/2305.16439, 2023. URL: https://arxiv.org/abs/2305.16439.
  26. Xian Li and Hongyu Gong. Robust optimization for multilingual translation with imbalanced data. In NeurIPS, pages 25086-25099, 2021. Google Scholar
  27. Natalia Martínez, Martín Bertrán, and Guillermo Sapiro. Minimax pareto fairness: A multi objective perspective. In ICML, volume 119 of Proceedings of Machine Learning Research, pages 6755-6764. PMLR, 2020. Google Scholar
  28. Fabrice Talla Nobibon and Roel Leus. Robust maximum weighted independent-set problems on interval graphs. Optim. Lett., 8(1):227-235, 2014. Google Scholar
  29. Christos H. Papadimitriou and Mihalis Yannakakis. On the approximability of trade-offs and optimal access of web sources. In FOCS, pages 86-92. IEEE Computer Society, 2000. Google Scholar
  30. Bozidar Radunovic and Jean-Yves Le Boudec. A unified framework for max-min and min-max fairness with applications. IEEE/ACM Trans. Netw., 15(5):1073-1083, 2007. Google Scholar
  31. Antonio Sedeño-Noda and Marcos Colebrook. A biobjective dijkstra algorithm. Eur. J. Oper. Res., 276(1):106-118, 2019. Google Scholar
  32. Paolo Serafini. Some considerations about computational complexity for multi objective combinatorial problems. In Recent Advances and Historical Development of Vector Optimization: Proceedings of an International Conference on Vector Optimization, pages 222-232. Springer, 1987. Google Scholar
  33. Jacobo Valdes, Robert Endre Tarjan, and Eugene L. Lawler. The recognition of series parallel digraphs. SIAM J. Comput., 11(2):298-313, 1982. Google Scholar