Fully Polynomial-time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909126990233600 |
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| author | Bentert, Matthias Heeger, Klaus Koana, Tomohiro |
| author_facet | Bentert, Matthias Heeger, Klaus Koana, Tomohiro |
| contents | 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 $ι' \le ι$ 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.
Alon and Yuster [ESA 2007] 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 computing the girth of a graph, randomized $O(ι^{ω- 1}n)$-time algorithms for Maximum Matching and for finding any induced four-vertex subgraph except for a clique or an independent set, and an $O(ι^{(ω-1)/2}n^2) \subseteq O(ι^{0.687} n^2)$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01839 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fully Polynomial-time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication Bentert, Matthias Heeger, Klaus Koana, Tomohiro Data Structures and Algorithms 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 $ι' \le ι$ 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. Alon and Yuster [ESA 2007] 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 computing the girth of a graph, randomized $O(ι^{ω- 1}n)$-time algorithms for Maximum Matching and for finding any induced four-vertex subgraph except for a clique or an independent set, and an $O(ι^{(ω-1)/2}n^2) \subseteq O(ι^{0.687} n^2)$-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. |
| title | Fully Polynomial-time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2403.01839 |