Submodular Norms with Applications To Online Facility Location and Stochastic Probing

Authors Kalen Patton, Matteo Russo, Sahil Singla

Thumbnail PDF


  • Filesize: 0.85 MB
  • 22 pages

Document Identifiers

Author Details

Kalen Patton
  • School of Mathematics, Georgia Tech, Atlanta, GA, USA
Matteo Russo
  • DIAG, Sapienza Università di Roma, Italy
Sahil Singla
  • School of Computer Science, Georgia Tech, Atlanta, GA, USA


We thank the anonymous reviewers of APPROX/RANDOM 2023 for their valuable feedback.

Cite AsGet BibTex

Kalen Patton, Matteo Russo, and Sahil Singla. Submodular Norms with Applications To Online Facility Location and Stochastic Probing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Optimization problems often involve vector norms, which has led to extensive research on developing algorithms that can handle objectives beyond 𝓁_p norms. Our work introduces the concept of submodular norms, which are a versatile type of norms that possess marginal properties similar to submodular set functions. We show that submodular norms can either accurately represent or approximate well-known classes of norms, such as 𝓁_p norms, ordered norms, and symmetric norms. Furthermore, we establish that submodular norms can be applied to optimization problems such as online facility location and stochastic probing. This allows us to develop a logarithmic-competitive algorithm for online facility location with symmetric norms, and to prove logarithmic adaptivity gap for stochastic probing with symmetric norms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Submodularity
  • Monotone Norms
  • Online Facility Location
  • Stochastic Probing


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Alexandr Andoni, Jaroslaw Blasiok, and Arnold Filtser. Communication complexity of inner product in symmetric normed spaces. In ITCS, volume 251, pages 4:1-4:22, 2023. Google Scholar
  2. Alexandr Andoni, Chengyu Lin, Ying Sheng, Peilin Zhong, and Ruiqi Zhong. Subspace embedding and linear regression with orlicz norm. In ICML, volume 80 of Proceedings of Machine Learning Research, pages 224-233. PMLR, 2018. Google Scholar
  3. Alexandr Andoni, Assaf Naor, Aleksandar Nikolov, Ilya P. Razenshteyn, and Erik Waingarten. Hölder homeomorphisms and approximate nearest neighbors. In FOCS, pages 159-169. IEEE Computer Society, 2018. Google Scholar
  4. Alexandr Andoni, Huy L. Nguyen, Aleksandar Nikolov, Ilya P. Razenshteyn, and Erik Waingarten. Approximate near neighbors for general symmetric norms. In STOC, pages 902-913. ACM, 2017. Google Scholar
  5. Arash Asadpour and Hamid Nazerzadeh. Maximizing stochastic monotone submodular functions. Manag. Sci., 62(8):2374-2391, 2016. Google Scholar
  6. Brian Axelrod, Yang P. Liu, and Aaron Sidford. Near-optimal approximate discrete and continuous submodular function minimization. In SODA, pages 837-853. SIAM, 2020. Google Scholar
  7. Yossi Azar, Niv Buchbinder, T.-H. Hubert Chan, Shahar Chen, Ilan Reuven Cohen, Anupam Gupta, Zhiyi Huang, Ning Kang, Viswanath Nagarajan, Joseph Naor, and Debmalya Panigrahi. Online algorithms for covering and packing problems with convex objectives. In FOCS, pages 148-157. IEEE Computer Society, 2016. Google Scholar
  8. Francis Bach. Submodular Functions: from Discrete to Continous Domains. Mathematical Programming, Series A, 2018. URL:
  9. Francis R. Bach. Structured sparsity-inducing norms through submodular functions. In Advances in Neural Information Processing Systems 23: 24th Annual Conference on Neural Information Processing Systems, pages 118-126, 2010. Google Scholar
  10. Francis R. Bach. Learning with submodular functions: A convex optimization perspective. Found. Trends Mach. Learn., 6(2-3):145-373, 2013. Google Scholar
  11. Rajendra Bhatia. Matrix Analysis, volume 169. Springer, 1997. Google Scholar
  12. Yatao Bian, Joachim M. Buhmann, and Andreas Krause. Continuous submodular function maximization. CoRR, abs/2006.13474, 2020. URL:
  13. Yatao An Bian, Joachim M. Buhmann, and Andreas Krause. Optimal continuous dr-submodular maximization and applications to provable mean field inference. In ICML, volume 97 of Proceedings of Machine Learning Research, pages 644-653. PMLR, 2019. Google Scholar
  14. Domagoj Bradac, Sahil Singla, and Goran Zuzic. (Near) optimal adaptivity gaps for stochastic multi-value probing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM, pages 49:1-49:21, 2019. Google Scholar
  15. Jaroslaw Byrka, Krzysztof Sornat, and Joachim Spoerhase. Constant-factor approximation for ordered k-median. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC, pages 620-631. ACM, 2018. Google Scholar
  16. Deeparnab Chakrabarty and Chaitanya Swamy. Approximation algorithms for minimum norm and ordered optimization problems. In STOC, pages 126-137. ACM, 2019. Google Scholar
  17. Deeparnab Chakrabarty and Chaitanya Swamy. Simpler and better algorithms for minimum-norm load balancing. In ESA, volume 144 of LIPIcs, pages 27:1-27:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  18. Shichuan Deng, Jian Li, and Yuval Rabani. Generalized unrelated machine scheduling problem. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, pages 2898-2916. SIAM, 2023. Google Scholar
  19. Hossein Esfandiari, Amin Karbasi, and Vahab S. Mirrokni. Adaptivity in adaptive submodularity. In Mikhail Belkin and Samory Kpotufe, editors, Conference on Learning Theory, COLT, volume 134, pages 1823-1846. PMLR, 2021. Google Scholar
  20. Moran Feldman and Amin Karbasi. Continuous submodular maximization: Beyond dr-submodularity. In NeurIPS, 2020. Google Scholar
  21. Dimitris Fotakis. On the competitive ratio for online facility location. Algorithmica, 50(1):1-57, 2008. Google Scholar
  22. Satoru Fujishige. Submodular functions and optimization. Elsevier, 2005. Google Scholar
  23. Anupam Gupta, Ravishankar Krishnaswamy, and Kirk Pruhs. Online primal-dual for non-linear optimization with applications to speed scaling. In Approximation and Online Algorithms - 10th International Workshop, WAOA, volume 7846, pages 173-186, 2012. Google Scholar
  24. Anupam Gupta and Viswanath Nagarajan. A stochastic probing problem with applications. In Integer Programming and Combinatorial Optimization - 16th International Conference, IPCO, pages 205-216, 2013. Google Scholar
  25. Anupam Gupta, Viswanath Nagarajan, and Sahil Singla. Algorithms and adaptivity gaps for stochastic probing. In Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1731-1747, 2016. Google Scholar
  26. Anupam Gupta, Viswanath Nagarajan, and Sahil Singla. Adaptivity gaps for stochastic probing: Submodular and XOS functions. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1688-1702. SIAM, 2017. Google Scholar
  27. Swati Gupta, Jai Moondra, and Mohit Singh. Socially fair and hierarchical facility location problems. In Proceedings of Economics and Computation (EC), 2023. Google Scholar
  28. Sharat Ibrahimpur. Stochastic minimum norm combinatorial optimization. Ph.D. Thesis, University of Waterloo,, 2022.
  29. Sharat Ibrahimpur and Chaitanya Swamy. Approximation algorithms for stochastic minimum-norm combinatorial optimization. In FOCS, pages 966-977. IEEE, 2020. Google Scholar
  30. Sharat Ibrahimpur and Chaitanya Swamy. Minimum-norm load balancing is (almost) as easy as minimizing makespan. In ICALP, volume 198, pages 81:1-81:20, 2021. Google Scholar
  31. Sharat Ibrahimpur and Chaitanya Swamy. A simple approximation algorithm for vector scheduling and applications to stochastic min-norm load balancing. In SOSA, pages 247-256. SIAM, 2022. Google Scholar
  32. Thomas Kesselheim, Marco Molinaro, and Sahil Singla. Online and bandit algorithms beyond 𝓁_p norms. In SODA, pages 1566-1593. SIAM, 2023. Google Scholar
  33. Thomas Kesselheim and Sahil Singla. Online learning with vector costs and bandits with knapsacks. In Proceedings of COLT, pages 2286-2305, 2020. Google Scholar
  34. Jerry Li, Aleksandar Nikolov, Ilya P. Razenshteyn, and Erik Waingarten. On mean estimation for general norms with statistical queries. In COLT, volume 99 of Proceedings of Machine Learning Research, pages 2158-2172. PMLR, 2019. Google Scholar
  35. Adam Meyerson. Online facility location. In FOCS, pages 426-431. IEEE Computer Society, 2001. Google Scholar
  36. Viswanath Nagarajan and Xiangkun Shen. Online covering with sum of 𝓁_q- norm objectives. In ICALP, volume 80 of LIPIcs, pages 12:1-12:12, 2017. Google Scholar
  37. Rad Niazadeh, Tim Roughgarden, and Joshua R. Wang. Optimal algorithms for continuous non-monotone submodular and dr-submodular maximization. In NeurIPS, pages 9617-9627, 2018. Google Scholar
  38. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science & Business Media, 2003. Google Scholar
  39. Zhao Song, Ruosong Wang, Lin F. Yang, Hongyang Zhang, and Peilin Zhong. Efficient symmetric norm regression via linear sketching. In NeurIPS, pages 828-838, 2019. Google Scholar
  40. Qixin Zhang, Zengde Deng, Zaiyi Chen, Haoyuan Hu, and Yu Yang. Stochastic continuous submodular maximization: Boosting via non-oblivious function. In ICML, volume 162 of Proceedings of Machine Learning Research, pages 26116-26134. PMLR, 2022. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail