Parametrized Complexity of Expansion Height

Authors Ulrich Bauer , Abhishek Rathod , Jonathan Spreer

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Author Details

Ulrich Bauer
  • Department of Mathematics, Technical University of Munich (TUM), Boltzmannstr. 3, 85748 Garching b. München, Germany
Abhishek Rathod
  • Department of Mathematics, Technical University of Munich (TUM), Boltzmannstr. 3, 85748 Garching b. München, Germany
Jonathan Spreer
  • School of Mathematics and Statistics, The University of Sydney, NSW 2006 Australia


We want to thank João Paixão for very helpful discussions.

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Ulrich Bauer, Abhishek Rathod, and Jonathan Spreer. Parametrized Complexity of Expansion Height. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Deciding whether two simplicial complexes are homotopy equivalent is a fundamental problem in topology, which is famously undecidable. There exists a combinatorial refinement of this concept, called simple-homotopy equivalence: two simplicial complexes are of the same simple-homotopy type if they can be transformed into each other by a sequence of two basic homotopy equivalences, an elementary collapse and its inverse, an elementary expansion. In this article we consider the following related problem: given a 2-dimensional simplicial complex, is there a simple-homotopy equivalence to a 1-dimensional simplicial complex using at most p expansions? We show that the problem, which we call the erasability expansion height, is W[P]-complete in the natural parameter p.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → W hierarchy
  • Simple-homotopy theory
  • simple-homotopy type
  • parametrized complexity theory
  • simplicial complexes
  • (modified) dunce hat


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