Online Covering with Sum of $ell_q$-Norm Objectives

Authors Viswanath Nagarajan, Xiangkun Shen

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Viswanath Nagarajan
Xiangkun Shen

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Viswanath Nagarajan and Xiangkun Shen. Online Covering with Sum of $ell_q$-Norm Objectives. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 12:1-12:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


We consider fractional online covering problems with lq-norm objectives. The problem of interest is of the form min{ f(x) : Ax >= 1, x >= 0} where f(x) is the weighted sum of lq-norms and A is a non-negative matrix. The rows of A (i.e. covering constraints) arrive online over time. We provide an online O(log d+log p)-competitive algorithm where p is the maximum to minimum ratio of A and A is the row sparsity of A. This is based on the online primal-dual framework where we use the dual of the above convex program. Our result expands the class of convex objectives that admit good online algorithms: prior results required a monotonicity condition on the objective which is not satisfied here. This result is nearly tight even for the linear special case. As direct applications, we obtain (i) improved online algorithms for non-uniform buy-at-bulk network design and (ii) the first online algorithm for throughput maximization under lq-norm edge capacities.
  • online algorithm
  • covering/packing problem
  • convex
  • buy-at-bulk
  • throughput maximization


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  1. Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. A general approach to online network optimization problems. ACM Transactions on Algorithms, 2(4):640-660, 2006. Google Scholar
  2. Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. The online set cover problem. SIAM J. Comput., 39(2):361-370, 2009. Google Scholar
  3. Baruch Awerbuch, Yossi Azar, Edward F. Grove, Ming-Yang Kao, P. Krishnan, and Jeffrey Scott Vitter. Load balancing in the l_p norm. In FOCS, pages 383-391, 1995. Google Scholar
  4. Baruch Awerbuch, Yossi Azar, and Serge Plotkin. Throughput-competitive on-line routing. In FOCS, pages 32-40. IEEE, 1993. Google Scholar
  5. Yossi Azar, Umang Bhaskar, Lisa K. Fleischer, and Debmalya Panigrahi. Online mixed packing and covering. In SODA, 2013. Google Scholar
  6. 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, 2016. Google Scholar
  7. Yossi Azar, Ilan Reuven Cohen, and Debmalya Panigrahi. Online covering with convex objectives and applications. CoRR, abs/1412.3507, 2014. URL:
  8. Nikhil Bansal, Niv Buchbinder, and Joseph Naor. Randomized competitive algorithms for generalized caching. SIAM J. Comput., 41(2):391-414, 2012. Google Scholar
  9. Nikhil Bansal and Kirk Pruhs. Server scheduling to balance priorities, fairness, and average quality of service. SIAM J. Comput., 39(7):3311-3335, 2010. Google Scholar
  10. Avrim Blum, Anupam Gupta, Yishay Mansour, and Ankit Sharma. Welfare and profit maximization with production costs. In FOCS, pages 77-86, 2011. Google Scholar
  11. Niv Buchbinder, Shahar Chen, Anupam Gupta, Viswanath Nagarajan, and Joseph Naor. Online packing and covering framework with convex objectives. CoRR, abs/1412.8347, 2014. Google Scholar
  12. Niv Buchbinder, Kamal Jain, and Joseph Naor. Online primal-dual algorithms for maximizing ad-auctions revenue. In ESA, pages 253-264, 2007. Google Scholar
  13. Niv Buchbinder and Joseph Naor. Online primal-dual algorithms for covering and packing. Math. Oper. Res., 34(2):270-286, 2009. Google Scholar
  14. Niv Buchbinder and Joseph (Seffi) Naor. The design of competitive online algorithms via a primal-dual approach. Found. Trends Theor. Comput. Sci., 3(2-3):93-263, 2007. Google Scholar
  15. T.-H. Hubert Chan, Zhiyi Huang, and Ning Kang. Online convex covering and packing problems. CoRR, abs/1502.01802, 2015. Google Scholar
  16. Moses Charikar and Adriana Karagiozova. On non-uniform multicommodity buy-at-bulk network design. In STOC, pages 176-182. ACM, 2005. Google Scholar
  17. Chandra Chekuri, Mohammad Taghi Hajiaghayi, Guy Kortsarz, and Mohammad R. Salavatipour. Approximation algorithms for nonuniform buy-at-bulk network design. SIAM J. Comput., 39(5):1772-1798, 2010. Google Scholar
  18. Nikhil R. Devanur and Zhiyi Huang. Primal dual gives almost optimal energy efficient online algorithms. In SODA, pages 1123-1140, 2014. Google Scholar
  19. Nikhil R. Devanur and Kamal Jain. Online matching with concave returns. In STOC, pages 137-144. ACM, 2012. Google Scholar
  20. Noa Elad, Satyen Kale, and Joseph (Seffi) Naor. Online semidefinite programming. In ICALP, pages 40:1-40:13, 2016. Google Scholar
  21. Alina Ene, Deeparnab Chakrabarty, Ravishankar Krishnaswamy, and Debmalya Panigrahi. Online buy-at-bulk network design. In FOCS, pages 545-562, 2015. Google Scholar
  22. Anupam Gupta, Ravishankar Krishnaswamy, and Kirk Pruhs. Online primal-dual for non-linear optimization with applications to speed scaling. In WAOA, pages 173-186, 2012. Google Scholar
  23. Anupam Gupta and Viswanath Nagarajan. Approximating sparse covering integer programs online. Math. Oper. Res., 39(4):998-1011, 2014. Google Scholar
  24. Zhiyi Huang and Anthony Kim. Welfare maximization with production costs: A primal dual approach. In SODA, pages 59-72, 2015. Google Scholar
  25. Ishai Menache and Mohit Singh. Online caching with convex costs: Extended abstract. In SPAA, pages 46-54, 2015. Google Scholar
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