Fine-grained Meta-Theorems for Vertex Integrity

TL;DR

This work studies FO and MSO model-checking parameterized by vertex integrity ${\iota(G)}$, an intermediate graph parameter between vertex cover and tree-depth. It presents kernelization-based meta-theorems: FO model-checking runs in ${2^{O(k^2 q + q\log q)} + n^{O(1)}}$ and MSO model-checking runs in ${2^{2^{O(k^2 + k q)}} + n^{O(1)}}$, with kernels of sizes ${O(2^{k^2} q k)}$ and ${O(2^{k^2 + k q} q_1 k)}$ respectively. The authors provide ETH-based lower bounds showing the $k^2$-level quadratic dependence on the parameter $k$ is necessary, placing vertex integrity in a truly intermediate position between vertex cover and tree-depth. These results refine our understanding of how expressive resources in FO/MSO interact with refined structural parameters and highlight potential avenues for extending to MSO$_2$ and related logics. The work informs both theory and the classification of parameterized graph problems by identifying precise complexity trade-offs for a natural intermediate graph parameter.

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 $φ$ with $q$ quantifiers, deciding if $G$ satisfies $φ$ 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.

Figures (2)

• Figure 1: The way the vertex set $S'$ intersects the vertex sets $C_1$ and $C_2$.
• Figure 2: The connection between $S$ and the set $W_{47}$. For this example $k=3$, we can represent up to $2^9$ numbers in binary. In order to represent $47_{10} = 000101111_2$, we shall connect $w_{(47,1)}$ with $s_4, s_5$ and $s_6$ in order to represent the three least significant bits (which are all 1), and $w_{(47,2)}$ with $s_4$ and $s_6$ to represent the next triad of bits. The three most significant bits are all 0, therefore $w_{(47,3)}$ is not connected to any of $s_4,s_5,s_6$.

Theorems & Definitions (24)

• Definition 2.1
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• Definition 2.5
• Definition 2.6
• Definition 2.8
• Lemma 2.9
• Theorem 3.1
• Theorem 3.2