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Lower Bounds for Ranking-Based Pivot Rules

Authors Yann Disser , Georg Loho , Matthew Maat , Nils Mosis

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LIPIcs.STACS.2026.31.pdf
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ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • lower bounds
  • Markov decision processes
  • parity games
  • pivot rules
  • policy iteration
  • simplex method

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Abstract

The existence of a polynomial pivot rule for the simplex method for linear programming, policy iteration for Markov decision processes, and strategy improvement for parity games each are prominent open problems in their respective fields. While numerous natural candidates for efficient rules have been eliminated, all existing lower bound constructions are tailored to individual or small sets of pivot rules. We introduce a unified framework for formalizing classes of rules according to the information about the input that they rely on. Within this framework, we show lower bounds for ranking-based classes of rules that base their decisions on orderings of the improving pivot steps induced by the underlying data. Our first result is a superpolynomial lower bound for strategy improvement, obtained via a family of sink parity games, which applies to memory-based generalizations of Bland’s rule that only access the input by comparing the ranks of improving edges in some global order. Our second result is a subexponential lower bound for policy iteration, obtained via a family of Markov decision processes, which applies to memoryless rules that only access the input by comparing improving actions according to their ranks in a global order, their reduced costs, and the associated improvements in objective value. Both results carry over to the simplex method for linear programming.

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Yann Disser, Georg Loho, Matthew Maat, and Nils Mosis. Lower Bounds for Ranking-Based Pivot Rules. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.31

Author Details

Yann Disser
  • TU Darmstadt, Germany
Georg Loho
  • FU Berlin, Germany
  • University of Twente, Enschede, The Netherlands
Matthew Maat
  • University of Twente, Enschede, The Netherlands
Nils Mosis
  • TU Darmstadt, Germany

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