Tight Running Time Lower Bounds for Vertex Deletion Problems

TL;DR

This paper establishes tight exponential time lower bounds for the Vertex Deletion problem across various graph classes and conditions, showing it cannot be solved faster than certain exponential thresholds under ETH.

Contribution

It adapts classical NP-hardness constructions to prove tight ETH-based lower bounds and provides a detailed complexity dichotomy for different graph classes and parameters.

Findings

No $2^{o(n)}$-time algorithm for $ ext{Pi}$-Vertex Deletion.

No $2^{o(n+m)}$-time algorithm if $ ext{Pi}$ contains all independent sets.

Certain cases allow algorithms with $2^{O( ext{sqrt}(m))}$ or $2^{o(m)}$ time.

Abstract

For a graph class $\Pi$, the $\Pi$-Vertex Deletion problem has as input an undirected graph $G=(V,E)$ and an integer $k$ and asks whether there is a set of at most $k$ vertices that can be deleted from $G$ such that the resulting graph is a member of $\Pi$. By a classic result of Lewis and Yannakakis [J. Comput. Syst. Sci. '80], $\Pi$-Vertex Deletion is NP-hard for all hereditary properties $\Pi$. We adapt the original NP-hardness construction to show that under the Exponential Time Hypothesis (ETH) tight complexity results can be obtained. We show that $\Pi$-Vertex Deletion does not admit a $2^{o(n)}$-time algorithm where $n$ is the number of vertices in $G$. We also obtain a dichotomy for running time bounds that include the number $m$ of edges in the input graph: On the one hand, if $\Pi$ contains all independent sets, then there is no $2^{o(n+m)}$-time algorithm for $\Pi$-Vertex Deletion. On the other hand, if there is a fixed independent set that is not contained in $\Pi$ and containment in $\Pi$ can determined in $2^{O(n)}$ time or $2^{o(m)}$ time, then $\Pi$-Vertex Deletion can be solved in $2^{O(\sqrt{m})}+O(n)$ or $2^{o({m})}+O(n)$ time, respectively. We also consider restrictions on the domain of the input graph $G$. For example, we obtain that $\Pi$-Vertex Deletion cannot be solved in $2^{o(\sqrt{n})}$ time if $G$ is planar and $\Pi$ is hereditary and contains and excludes infinitely many planar graphs. Finally, we provide similar results for the problem variant where the deleted vertex set has to induce a connected graph.