Parameterized Lower Bounds and Dichotomy Results for the NP-completeness of $H$-free Edge Modification Problems

TL;DR

This paper classifies the NP-completeness of $H$-free edge modification problems, establishing precise conditions for complexity and showing they cannot be solved in subexponential parameterized time under ETH.

Contribution

It provides a complete complexity classification for $H$-free edge modification problems and proves their resistance to subexponential algorithms under ETH.

Findings

NP-complete if $H$ has at least two edges (deletion)

NP-complete if $H$ has at least two non-edges (completion)

NP-complete if $H$ has at least three vertices (editing)

Abstract

For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. $H$-free Edge Completion and $H$-free Edge Editing are defined similarly where only completion (addition) of edges are allowed in the former and both completion and deletion are allowed in the latter. We completely settle the classical complexities of these problems by proving that $H$-free Edge Deletion is NP-complete if and only if $H$ is a graph with at least two edges, $H$-free Edge Completion is NP-complete if and only if $H$ is a graph with at least two non-edges and $H$-free Edge Editing is NP-complete if and only if $H$ is a graph with at least three vertices. Additionally, we prove that, these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cdot |G|^{O(1)}$, unless Exponential Time Hypothesis fails. Furthermore, we obtain implications on the incompressibility of these problems.