Fully Polynomial-Time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication

Authors Matthias Bentert, Klaus Heeger, Tomohiro Koana

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Author Details

Matthias Bentert
  • University of Bergen, Norway
Klaus Heeger
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Tomohiro Koana
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany


We thank Vincent Borko for fruitful discussions regarding the results for finding small induced subgraphs and anonymous reviewers for their constructive feedback which, in particular, helped improving the running time of our algorithm for All-Pairs Shortest Paths.

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Matthias Bentert, Klaus Heeger, and Tomohiro Koana. Fully Polynomial-Time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study the computational complexity of several polynomial-time-solvable graph problems parameterized by vertex integrity, a measure of a graph’s vulnerability to vertex removal in terms of connectivity. Vertex integrity is the smallest number ι such that there is a set S of ι' ≤ ι vertices such that every connected component of G-S contains at most ι-ι' vertices. It is known that the vertex integrity lies between the well-studied parameters vertex cover number and tree-depth. Our work follows similar studies for vertex cover number [Alon and Yuster, ESA 2007] and tree-depth [Iwata, Ogasawara, and Ohsaka, STACS 2018]. Alon and Yuster designed algorithms for graphs with small vertex cover number using fast matrix multiplications. We demonstrate that fast matrix multiplication can also be effectively used when parameterizing by vertex integrity ι by developing efficient algorithms for problems including an O(ι^{ω-1}n)-time algorithm for Maximum Matching and an O(ι^{(ω-1)/2}n²) ⊆ O(ι^{0.687} n²)-time algorithm for All-Pairs Shortest Paths. These algorithms can be faster than previous algorithms parameterized by tree-depth, for which fast matrix multiplication is not known to be effective.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Matchings and factors
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Shortest paths
  • FPT in P
  • Algebraic Algorithms
  • Adaptive Algorithms
  • Subgraph Detection
  • Matching
  • APSP


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