Search-Space Reduction via Essential Vertices

Authors Benjamin Merlin Bumpus , Bart M. P. Jansen , Jari J. H. de Kroon

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Benjamin Merlin Bumpus
  • Eindhoven University of Technology, The Netherlands
Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Jari J. H. de Kroon
  • Eindhoven University of Technology, The Netherlands

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Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jari J. H. de Kroon. Search-Space Reduction via Essential Vertices. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Packing and covering problems
  • Theory of computation → Linear programming
  • Theory of computation → Fixed parameter tractability
  • fixed-parameter tractability
  • essential vertices
  • covering versus packing


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