Fully Polynomial-time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication
ESA(2024)
摘要
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.
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-1n)-time algorithm for computing the girth of a graph,
randomized O(ι^ω - 1n)-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)/2n^2) ⊆ 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.
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