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A Unified Worst Case for Classical Simplex and Policy Iteration Pivot Rules

Authors Yann Disser , Nils Mosis

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  • 17 pages

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Yann Disser
  • TU Darmstadt, Germany
Nils Mosis
  • TU Darmstadt, Germany

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Yann Disser and Nils Mosis. A Unified Worst Case for Classical Simplex and Policy Iteration Pivot Rules. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 27:1-27:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We construct a family of Markov decision processes for which the policy iteration algorithm needs an exponential number of improving switches with Dantzig’s rule, with Bland’s rule, and with the Largest Increase pivot rule. This immediately translates to a family of linear programs for which the simplex algorithm needs an exponential number of pivot steps with the same three pivot rules. Our results yield a unified construction that simultaneously reproduces well-known lower bounds for these classical pivot rules, and we are able to infer that any (deterministic or randomized) combination of them cannot avoid an exponential worst-case behavior. Regarding the policy iteration algorithm, pivot rules typically switch multiple edges simultaneously and our lower bound for Dantzig’s rule and the Largest Increase rule, which perform only single switches, seem novel. Regarding the simplex algorithm, the individual lower bounds were previously obtained separately via deformed hypercube constructions. In contrast to previous bounds for the simplex algorithm via Markov decision processes, our rigorous analysis is reasonably concise.

Subject Classification

ACM Subject Classification
  • Theory of computation → Linear programming
  • Mathematics of computing → Markov processes
  • Bland’s pivot rule
  • Dantzig’s pivot rule
  • Largest Increase pivot rule
  • Markov decision process
  • policy iteration
  • simplex algorithm


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  1. Ilan Adler, Richard M Karp, and Ron Shamir. A simplex variant solving an m× d linear program in o (min (m2, d2) expected number of pivot steps. Journal of Complexity, 3(4):372-387, 1987. Google Scholar
  2. Ilan Adler, Christos Papadimitriou, and Aviad Rubinstein. On simplex pivoting rules and complexity theory. In Integer Programming and Combinatorial Optimization: 17th International Conference, IPCO 2014., pages 13-24. Springer, 2014. Google Scholar
  3. Nina Amenta and Günter M Ziegler. Deformed products and maximal shadows of polytopes. Contemporary Mathematics, 223:57-90, 1999. Google Scholar
  4. David Avis and Vasek Chvátal. Notes on Bland’s pivoting rule. Polyhedral Combinatorics: Dedicated to the memory of D.R. Fulkerson, pages 24-34, 1978. Google Scholar
  5. David Avis and Oliver Friedmann. An exponential lower bound for Cunningham’s rule. Mathematical Programming, 161:271-305, 2017. Google Scholar
  6. Richard Bellman. Dynamic Programming. Princeton University Press, 1957. Google Scholar
  7. Dimitris Bertsimas and Santosh Vempala. Solving convex programs by random walks. Journal of the ACM, 51(4):540-556, 2004. Google Scholar
  8. Robert G Bland. New finite pivoting rules for the simplex method. Mathematics of Operations Research, 2(2):103-107, 1977. Google Scholar
  9. Karl-Heinz Borgwardt. The average number of pivot steps required by the simplex-method is polynomial. Zeitschrift für Operations Research, 26:157-177, 1982. Google Scholar
  10. Daniel Dadush and Sophie Huiberts. A friendly smoothed analysis of the simplex method. In Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC), pages 390-403, 2018. Google Scholar
  11. George Dantzig. Linear programming and extensions. Princeton university press, 1963. Google Scholar
  12. George B. Dantzig. Maximization of a linear function of variables subject to linear inequalities. Activity analysis of production and allocation, 13:339-347, 1951. Google Scholar
  13. George B. Dantzig, Alex Orden, and Philip Wolfe. The generalized simplex method for minimizing a linear form under linear inequality restraints. Pacific Journal of Mathematics, 5(2):183-195, 1955. Google Scholar
  14. Amit Deshpande and Daniel A Spielman. Improved smoothed analysis of the shadow vertex simplex method. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pages 349-356. IEEE, 2005. Google Scholar
  15. Yann Disser, Oliver Friedmann, and Alexander V. Hopp. An exponential lower bound for zadeh’s pivot rule. Mathematical Programming, 199(1-2):865-936, 2023. Google Scholar
  16. Yann Disser and Martin Skutella. The simplex algorithm is NP-mighty. ACM Transactions on Algorithms (TALG), 15(1):1-19, 2018. Google Scholar
  17. John Dunagan and Santosh Vempala. A simple polynomial-time rescaling algorithm for solving linear programs. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pages 315-320, 2004. Google Scholar
  18. John Fearnley. Exponential lower bounds for policy iteration. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP), pages 551-562, 2010. Google Scholar
  19. John Fearnley and Rahul Savani. The complexity of the simplex method. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC), pages 201-208, 2015. Google Scholar
  20. Oliver Friedmann. Exponential lower bounds for solving infinitary payoff games and linear programs. PhD thesis, LMU Munich, 2011. Google Scholar
  21. Oliver Friedmann, Thomas D. Hansen, and Uri Zwick. Subexponential lower bounds for randomized pivoting rules for the simplex algorithm. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC), pages 283-292, 2011. Google Scholar
  22. Bernd Gärtner. The random-facet simplex algorithm on combinatorial cubes. Random Structures & Algorithms, 20(3):353-381, 2002. Google Scholar
  23. Bernd Gärtner and Ingo Schurr. Linear programming and unique sink orientations. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 749-757, 2006. Google Scholar
  24. Saul Gass and Thomas Saaty. The computational algorithm for the parametric objective function. Naval research logistics quarterly, 2(1-2):39-45, 1955. Google Scholar
  25. Donald Goldfarb and William Y. Sit. Worst case behavior of the steepest edge simplex method. Discrete Applied Mathematics, 1(4):277-285, 1979. Google Scholar
  26. Thomas D. Hansen. Worst-case analysis of strategy iteration and the simplex method. PhD thesis, Aarhus University, 2012. Google Scholar
  27. Thomas D. Hansen and Uri Zwick. An improved version of the random-facet pivoting rule for the simplex algorithm. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC), pages 209-218, 2015. Google Scholar
  28. Ronald A. Howard. Dynamic programming and Markov processes. John Wiley, 1960. Google Scholar
  29. Sophie Huiberts, Yin Tat Lee, and Xinzhi Zhang. Upper and lower bounds on the smoothed complexity of the simplex method. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC), pages 1904-1917, 2023. Google Scholar
  30. Robert G. Jeroslow. The simplex algorithm with the pivot rule of maximizing criterion improvement. Discrete Mathematics, 4(4):367-377, 1973. Google Scholar
  31. Gil Kalai. A subexponential randomized simplex algorithm. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing (STOC), pages 475-482, 1992. Google Scholar
  32. Narendra Karmarkar. A new polynomial-time algorithm for linear programming. In Proceedings of the 16th Annual ACM Symposium on Theory of Computing (STOC), pages 302-311, 1984. Google Scholar
  33. Jonathan A Kelner and Daniel A Spielman. A randomized polynomial-time simplex algorithm for linear programming. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 51-60, 2006. Google Scholar
  34. Leonid G. Khachiyan. Polynomial algorithms in linear programming. USSR Computational Mathematics and Mathematical Physics, 20(1):53-72, 1980. Google Scholar
  35. Victor Klee and George J Minty. How good is the simplex algorithm? Inequalities, 3(3):159-175, 1972. Google Scholar
  36. Jiří Matoušek, Micha Sharir, and Emo Welzl. A subexponential bound for linear programming. Algorithmica, 16(4/5):498-516, 1996. Google Scholar
  37. Mary Melekopoglou and Anne Condon. On the complexity of the policy improvement algorithm for Markov decision processes. ORSA Journal on Computing, 6(2):188-192, 1994. Google Scholar
  38. Katta G. Murty. Computational complexity of parametric linear programming. Mathematical Programming, 19(1):213-219, 1980. Google Scholar
  39. Martin L. Puterman. Markov decision processes: discrete stochastic dynamic programming. John Wiley & Sons, 1994. Google Scholar
  40. Ingo Schurr and Tibor Szabó. Finding the sink takes some time: An almost quadratic lower bound for finding the sink of unique sink oriented cubes. Discrete & Computational Geometry, 31(4):627-642, 2004. Google Scholar
  41. Steve Smale. Mathematical problems for the next century. Mathematics: frontiers and perspectives, pages 271-294, 2000. Google Scholar
  42. Daniel A. Spielman and Shang-Hua Teng. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. Journal of the ACM, 51(3):385-463, 2004. Google Scholar
  43. Tibor Szabó and Emo Welzl. Unique sink orientations of cubes. In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science (FOCS), pages 547-555, 2001. Google Scholar
  44. Michael J Todd. Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems. Mathematical Programming, 35(2):173-192, 1986. Google Scholar
  45. Roman Vershynin. Beyond hirsch conjecture: walks on random polytopes and smoothed complexity of the simplex method. SIAM Journal on Computing, 39(2):646-678, 2009. Google Scholar
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