Fine-grained Meta-Theorems for Vertex Integrity

TL;DR

This paper establishes new algorithmic meta-theorems for vertex integrity, showing that FO and MSO model checking can be done efficiently with kernelization, and proves that the quadratic dependence on vertex integrity is optimal under ETH.

Contribution

It introduces fine-grained algorithmic meta-theorems for vertex integrity, providing complexity bounds and lower bounds that position it between vertex cover and tree-depth.

Findings

Model checking for FO logic runs in 2^{O(k^2q+q log q)} time.

Model checking for MSO logic runs in 2^{2^{O(k^2+kq)}} time.

Quadratic dependence on vertex integrity is proven to be optimal under ETH.

Abstract

Vertex Integrity is a graph measure which sits squarely between two more well-studied notions, namely vertex cover and tree-depth, and that has recently gained attention as a structural graph parameter. In this paper we investigate the algorithmic trade-offs involved with this parameter from the point of view of algorithmic meta-theorems for First-Order (FO) and Monadic Second Order (MSO) logic. Our positive results are the following: (i) given a graph $G$ of vertex integrity $k$ and an FO formula $\phi$ with $q$ quantifiers, deciding if $G$ satisfies $\phi$ can be done in time $2^{O(k^2q+q\log q)}+n^{O(1)}$; (ii) for MSO formulas with $q$ quantifiers, the same can be done in time $2^{2^{O(k^2+kq)}}+n^{O(1)}$. Both results are obtained using kernelization arguments, which pre-process the input to sizes $2^{O(k^2)}q$ and $2^{O(k^2+kq)}$ respectively. The complexities of our meta-theorems are significantly better than the corresponding meta-theorems for tree-depth, which involve towers of exponentials. However, they are worse than the roughly $2^{O(kq)}$ and $2^{2^{O(k+q)}}$ complexities known for corresponding meta-theorems for vertex cover. To explain this deterioration we present two formula constructions which lead to fine-grained complexity lower bounds and establish that the dependence of our meta-theorems on $k$ is the best possible. More precisely, we show that it is not possible to decide FO formulas with $q$ quantifiers in time $2^{o(k^2q)}$, and that there exists a constant-size MSO formula which cannot be decided in time $2^{2^{o(k^2)}}$, both under the ETH. Hence, the quadratic blow-up in the dependence on $k$ is unavoidable and vertex integrity has a complexity for FO and MSO logic which is truly intermediate between vertex cover and tree-depth.