@inproceedings{390964189244407581117acf7ad5180a,

title = "Fully Polynomial-Time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication",

abstract = "We study the computational complexity of several polynomial-time-solvable graph problems parameterized by vertex integrity, a measure of a graph{\textquoteright}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(ιω−1n)-time algorithm for Maximum Matching and an O(ι(ω−1)/2n2) ⊆ O(ι0.687n2)-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.",

keywords = "Adaptive Algorithms, Algebraic Algorithms, APSP, FPT in P, Matching, Subgraph Detection",

author = "Matthias Bentert and Klaus Heeger and Tomohiro Koana",

note = "Publisher Copyright: {\textcopyright} Matthias Bentert, Klaus Heeger, and Tomohiro Koana.; 31st Annual European Symposium on Algorithms, ESA 2023 ; Conference date: 04-09-2023 Through 06-09-2023",

year = "2023",

month = sep,

day = "1",

doi = "10.4230/LIPIcs.ESA.2023.16",

language = "English",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

editor = "{Li Gortz}, Inge and Martin Farach-Colton and Puglisi, {Simon J.} and Grzegorz Herman",

booktitle = "31st Annual European Symposium on Algorithms, ESA 2023",

address = "Germany",

}