Mathematical Programming

, Volume 175, Issue 1–2, pp 307–353 | Cite as

Nearly linear-time packing and covering LP solvers

Achieving width-independence and -convergence
  • Zeyuan Allen-ZhuEmail author
  • Lorenzo Orecchia
Full Length Paper Series A


Packing and covering linear programs (PC-LP s) form an important class of linear programs (LPs) across computer science, operations research, and optimization. Luby and Nisan (in: STOC, ACM Press, New York, 1993) constructed an iterative algorithm for approximately solving PC-LP s in nearly linear time, where the time complexity scales nearly linearly in N, the number of nonzero entries of the matrix, and polynomially in \(\varepsilon \), the (multiplicative) approximation error. Unfortunately, existing nearly linear-time algorithms (Plotkin et al. in Math Oper Res 20(2):257–301, 1995; Bartal et al., in: Proceedings 38th annual symposium on foundations of computer science, IEEE Computer Society, 1997; Young, in: 42nd annual IEEE symposium on foundations of computer science (FOCS’01), IEEE Computer Society, 2001; Koufogiannakis and Young in Algorithmica 70:494–506, 2013; Young in Nearly linear-time approximation schemes for mixed packing/covering and facility-location linear programs, 2014. arXiv:1407.3015; Allen-Zhu and Orecchia, in: SODA, 2015) for solving PC-LP s require time at least proportional to \(\varepsilon ^{-2}\). In this paper, we break this longstanding barrier by designing a packing solver that runs in time \(\widetilde{O}(N \varepsilon ^{-1})\) and covering LP solver that runs in time \(\widetilde{O}(N \varepsilon ^{-1.5})\). Our packing solver can be extended to run in time \(\widetilde{O}(N \varepsilon ^{-1})\) for a class of well-behaved covering programs. In a follow-up work, Wang et al. (in: ICALP, 2016) showed that all covering LPs can be converted into well-behaved ones by a reduction that blows up the problem size only logarithmically.

Mathematics Subject Classification

90C05, Linear programming 90C25, Convex programming 65K05, Mathematical programming methods 49M20, Methods of relaxation type 


  1. 1.
    Allen-Zhu, Z., Lee, Y.T., Orecchia, L.: Using optimization to obtain a width-independent, parallel, simpler, and faster positive SDP solver. In: SODA (2016)Google Scholar
  2. 2.
    Allen-Zhu, Z., Li, Y., Oliveira, R., Wigderson, A: Much faster algorithms for matrix scaling. In: FOCS, 2017. arXiv:1704.02315
  3. 3.
    Allen-Zhu, Z., Liao, Z., Orecchia, L.: Spectral sparsification and regret minimization beyond multiplicative updates. In: STOC (2015)Google Scholar
  4. 4.
    Allen-Zhu, Z., Orecchia, L.: Using optimization to break the epsilon barrier: a faster and simpler width-independent algorithm for solving positive linear programs in parallel. In: SODA (2015)Google Scholar
  5. 5.
    Allen-Zhu, Z., Orecchia, L.: Linear coupling: an ultimate unification of gradient and mirror descent. In: ITCS (2017)Google Scholar
  6. 6.
    Allen-Zhu, Z., Qu, Z., Richtárik, P., Yuan, Y.: Even faster accelerated coordinate descent using non-uniform sampling. In: ICML (2016)Google Scholar
  7. 7.
    Arora, S., Hazan, E., Kale, S.: The multiplicative weights update method: a meta-algorithm and applications. Theory Comput. 8, 121–164 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Awerbuch, B., Khandekar, R.: Stateless distributed gradient descent for positive linear programs. In: STOC (2008)Google Scholar
  9. 9.
    Awerbuch, B., Khandekar, R., Rao, S.: Distributed algorithms for multicommodity flow problems via approximate steepest descent framework. ACM Trans. Algorithms 9(1), 1–14 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bartal, Y., Byers, J.W., Raz, D.: Global optimization using local information with applications to flow control. In: Proceedings 38th Annual Symposium on Foundations of Computer Science, pp. 303–312. IEEE Computer Society (1997)Google Scholar
  11. 11.
    Bartal, Y., Byers, J.W., Raz, D.: Fast, distributed approximation algorithms for positive linear programming with applications to flow control. SIAM J. Comput. 33(6), 1261–1279 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ben-Tal, A., Arkadi, N.: Lectures on modern convex optimization. Soc. Ind. Appl. Math. 315–341 (2013)Google Scholar
  13. 13.
    Bienstock, D., Iyengar, G.: Faster approximation algorithms for packing and covering problems. Technical report, Columbia University, September 2004. Preliminary version published in STOC ’04Google Scholar
  14. 14.
    Byers, J., Nasser, G.: Utility-based decision-making in wireless sensor networks. In: 2000 first annual workshop on mobile and ad hoc networking and computing, 2000. MobiHOC, pp. 143–144. IEEE (2000)Google Scholar
  15. 15.
    Chudak, F.A., Eleutério, V. : Improved approximation schemes for linear programming relaxations of combinatorial optimization problems. In: Proceedings of the 11th International IPCO Conference on Integer Programming and Combinatorial Optimization, pp. 81–96 (2005)Google Scholar
  16. 16.
    Duan, R., Pettie, S.: Linear-time approximation for maximum weight matching. J. ACM 61(1), 1–23 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fercoq, O., Richtárik, P.: Accelerated, parallel and proximal coordinate descent. SIAM J. Optim. 25(4), 1997–2023 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fleischer, L.K.: Approximating fractional multicommodity flow independent of the number of commodities. SIAM J. Discrete Math. 13(4), 505–520 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Garg, N., Könemann, J.: Faster and simpler algorithms for multicommodity flow and other fractional packing problems. SIAM J. Comput. 37(2), 630–652 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Grigoriadis, M.D., Khachiyan, L.G.: Fast approximation schemes for convex programs with many blocks and coupling constraints. SIAM J. Optim. 4(1), 86–107 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jain, R., Ji, Z., Upadhyay, S., Watrous, J.: QIP = PSPACE. J. ACM JACM 58(6), 30 (2011)MathSciNetGoogle Scholar
  22. 22.
    Klein, P., Young, N.E.: On the number of iterations for Dantzig–Wolfe optimization and packing-covering approximation algorithms. SIAM J. Comput. 44(4), 1154–1172 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Koufogiannakis, C., Young, N.E.: A nearly linear-time PTAS for explicit fractional packing and covering linear programs. Algorithmica 70, 494–506 (2013). (Previously appeared in FOCS ’07) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Luby, M., Nisan, N.: A parallel approximation algorithm for positive linear programming. In: STOC, pp. 448–457. ACM Press, New York (1993)Google Scholar
  25. 25.
    Madry, A.: Faster approximation schemes for fractional multicommodity flow problems via dynamic graph algorithms. In: STOC. ACM Press, New York (2010)Google Scholar
  26. 26.
    Nemirovski, A.: Prox-method with rate of convergence \(O(1/t)\) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM J. Optim. 15(1), 229–251 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nesterov, Y.: Rounding of convex sets and efficient gradient methods for linear programming problems. Optim. Methods Softw. 23(1), 109–128 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Nesterov, Y.: A method of solving a convex programming problem with convergence rate \(O(1/k^2)\). Sov. Math. Dokl. 269, 543–547 (1983)Google Scholar
  29. 29.
    Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. 103(1), 127–152 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nesterov, Y.: Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM J Optim. 22(2), 341–362 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Plotkin, S.A., Shmoys, D.B., Tardos, É.: Fast approximation algorithms for fractional packing and covering problems. Math. Oper. Res. 20(2), 257–301 (1995). (conference version published in FOCS 1991) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Trevisan, L.: Parallel approximation algorithms by positive linear programming. Algorithmica 21(1), 72–88 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang, D., Mahoney, M., Mohan, N., Rao, S.: Faster parallel solver for positive linear programs via dynamically-bucketed selective coordinate descent. ArXiv e-prints arXiv:1511.06468 (2015)
  34. 34.
    Wang, D., Rao, S., Mahoney, M.W.: Unified acceleration method for packing and covering problems via diameter reduction. In: ICALP (2016)Google Scholar
  35. 35.
    Young, N.E.: Sequential and parallel algorithms for mixed packing and covering. In: 42nd Annual IEEE Symposium on Foundations of Computer Science (FOCS’01), pp. 538–546. IEEE Computer Society (2001)Google Scholar
  36. 36.
    Young, N.E.: Nearly linear-time approximation schemes for mixed packing/covering and facility-location linear programs. ArXiv e-prints arXiv:1407.3015 (2014)
  37. 37.
    Zurel, E., Nisan, N.: An efficient approximate allocation algorithm for combinatorial auctions. In: Proceedings of the 3rd ACM Conference on Electronic Commerce, pp. 125–136. ACM (2001)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Microsoft Research AIRedmondUSA
  2. 2.Boston UniversityBostonUSA

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