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

Authors Daniel Dadush , Zhuan Khye Koh , Bento Natura , László A. Végh



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Author Details

Daniel Dadush
  • CWI, Amsterdam, The Netherlands
Zhuan Khye Koh
  • Department of Mathematics, London School of Economics and Political Science, UK
Bento Natura
  • Department of Mathematics, London School of Economics and Political Science, UK
László A. Végh
  • Department of Mathematics, London School of Economics and Political Science, UK

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 AsGet BibTex

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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