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# An Accelerated Newton-Dinkelbach Method and Its Application to Two Variables per Inequality Systems

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LIPIcs.ESA.2021.36.pdf
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## Acknowledgements

The fourth author would like to thank Neil Olver for several inspiring discussions on 2VPI systems, in particular, on symmetries of the problem.

## Cite As

Daniel Dadush, Zhuan Khye Koh, Bento Natura, and László A. Végh. An Accelerated Newton-Dinkelbach Method and Its Application to Two Variables per Inequality Systems. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.36

## Abstract

We present an accelerated, or "look-ahead" version of the Newton-Dinkelbach method, a well-known technique for solving fractional and parametric optimization problems. This acceleration halves the Bregman divergence between the current iterate and the optimal solution within every two iterations. Using the Bregman divergence as a potential in conjunction with combinatorial arguments, we obtain strongly polynomial algorithms in three applications domains: (i) For linear fractional combinatorial optimization, we show a convergence bound of O(mlog m) iterations; the previous best bound was O(m²log m) by Wang et al. (2006). (ii) We obtain a strongly polynomial label-correcting algorithm for solving linear feasibility systems with two variables per inequality (2VPI). For a 2VPI system with n variables and m constraints, our algorithm runs in O(mn) iterations. Every iteration takes O(mn) time for general 2VPI systems, and O(m + nlog n) time for the special case of deterministic Markov Decision Processes (DMDPs). This extends and strengthens a previous result by Madani (2002) that showed a weakly polynomial bound for a variant of the Newton–Dinkelbach method for solving DMDPs. (iii) We give a simplified variant of the parametric submodular function minimization result by Goemans et al. (2017).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Mathematics of computing → Mathematical optimization
##### Keywords
• Newton-Dinkelbach method
• fractional optimization
• parametric optimization
• strongly polynomial algorithms
• two variables per inequality systems
• Markov decision processes
• submodular function minimization

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## References

1. Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network Flows - Theory, Algorithms and Applications. Prentice Hall, 1993.
2. Bengt Aspvall and Yossi Shiloach. A polynomial time algorithm for solving systems of linear inequalities with two variables per inequality. SIAM J. Comput., 9(4):827-845, 1980.
3. Edith Cohen and Nimrod Megiddo. Improved algorithms for linear inequalities with two variables per inequality. SIAM J. Comput., 23(6):1313-1347, 1994.
4. Werner Dinkelbach. On nonlinear fractional programming. Management Science, 13(7):492-498, 1967.
5. Herbert Edelsbrunner, Günter Rote, and Emo Welzl. Testing the necklace condition for shortest tours and optimal factors in the plane. Theor. Comput. Sci., 66(2):157-180, 1989.
6. Eugene A. Feinberg and Jefferson Huang. The value iteration algorithm is not strongly polynomial for discounted dynamic programming. Oper. Res. Lett., 42(2):130-131, 2014.
7. Michel X. Goemans, Swati Gupta, and Patrick Jaillet. Discrete Newton’s algorithm for parametric submodular function minimization. In Proceedings of the 19th International Conference on Integer Programming and Combinatorial Optimization, pages 212-227, 2017.
8. Andrew V. Goldberg and Robert E. Tarjan. Finding minimum-cost circulations by canceling negative cycles. J. ACM, 36(4):873-886, 1989.
9. Thomas Dueholm Hansen, Haim Kaplan, and Uri Zwick. Dantzig’s pivoting rule for shortest paths, deterministic MDPs, and minimum cost to time ratio cycles. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 847-860, 2014.
10. Dorit S. Hochbaum, Nimrod Megiddo, Joseph Naor, and Arie Tamir. Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality. Math. Program., 62:69-83, 1993.
11. Dorit S. Hochbaum and Joseph Naor. Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM J. Comput., 23(6):1179-1192, 1994.
12. Michael L. Littman, Thomas L. Dean, and Leslie Pack Kaelbling. On the complexity of solving Markov decision problems. In Proceedings of the 11th Annual Conference on Uncertainty in Artificial Intelligence, pages 394-402, 1995.
13. Omid Madani. On policy iteration as a Newton’s method and polynomial policy iteration algorithms. In Proceedings of the 18th National Conference on Artificial Intelligence, pages 273-278, 2002.
14. Nimrod Megiddo. Combinatorial optimization with rational objective functions. Math. Oper. Res., 4(4):414-424, 1979.
15. Nimrod Megiddo. Towards a genuinely polynomial algorithm for linear programming. SIAM J. Comput., 12(2):347-353, 1983.
16. Neil Olver and László A. Végh. A simpler and faster strongly polynomial algorithm for generalized flow maximization. J. ACM, 67(2):10:1-10:26, 2020.
17. Ian Post and Yinyu Ye. The simplex method is strongly polynomial for deterministic Markov decision processes. Math. Oper. Res., 40(4):859-868, 2015.
18. Tomasz Radzik. Newton’s method for fractional combinatorial optimization. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pages 659-669, 1992.
19. Tomasz Radzik. Fractional combinatorial optimization. In Ding-Zhu Du and Panos M. Pardalos, editors, Handbook of Combinatorial Optimization: Volume 1-3, pages 429-478. Springer US, 1998.
20. Tomasz Radzik and Andrew V. Goldberg. Tight bounds on the number of minimum-mean cycle cancellations and related results. Algorithmica, 11(3):226-242, 1994.
21. Robert E. Shostak. Deciding linear inequalities by computing loop residues. J. ACM, 28(4):769-779, 1981.
22. Steve Smale. Mathematical problems for the next century. The Mathematical Intelligencer, 20:7-15, 1998.
23. Éva Tardos. A strongly polynomial minimum cost circulation algorithm. Combinatorica, 5(3):247-256, 1985.
24. László A. Végh. A strongly polynomial algorithm for generalized flow maximization. Math. Oper. Res., 42(1):179-211, 2017.
25. Qin Wang, Xiaoguang Yang, and Jianzhong Zhang. A class of inverse dominant problems under weighted 𝓁_∞ norm and an improved complexity bound for Radzik’s algorithm. J. Global Optimization, 34(4):551-567, 2006.
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