Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations

Authors Joel D. Day, Florin Manea, Dirk Nowotka

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Author Details

Joel D. Day
  • Loughborough University, UK
  • Kiel University, Germany
Florin Manea
  • Kiel University, Germany
Dirk Nowotka
  • Kiel University, Germany

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Joel D. Day, Florin Manea, and Dirk Nowotka. Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


It is a long standing conjecture that the problem of deciding whether a quadratic word equation has a solution is in NP. It has also been conjectured that the length of a minimal solution to a quadratic equation is at most exponential in the length of the equation, with the latter conjecture implying the former. We show that both conjectures hold for some natural subclasses of quadratic equations, namely the classes of regular-reversed, k-ordered, and variable-sparse quadratic equations. We also discuss a connection of our techniques to the topic of unavoidable patterns, and the possibility of exploiting this connection to produce further similar results.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
  • Theory of computation → Problems, reductions and completeness
  • Quadratic Word Equations
  • Length Upper Bounds
  • NP
  • Unavoidable Patterns


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