Parameterized complexity dichotomy for $(r,\ell)$-Vertex Deletion

TL;DR

This paper establishes fixed-parameter tractability and single-exponential algorithms for the parameterized $(r,\, ext{ extlangle} \, ext{ell} ext{ extgreater})$-Vertex Deletion problem, resolving previously unknown cases and providing a complexity dichotomy.

Contribution

It proves that the cases $(2,1)$, $(1,2)$, and $(2,2)$ are fixed-parameter tractable with optimal single-exponential algorithms, and extends the analysis to a variant with independent set constraints.

Findings

All three previously unresolved cases are FPT.

Algorithms run in single-exponential time in parameter $k$.

Provides a complexity dichotomy for the problem variants.

Abstract

For two integers $r, \ell \geq 0$, a graph $G = (V, E)$ is an $(r,\ell)$-graph if $V$ can be partitioned into $r$ independent sets and $\ell$ cliques. In the parameterized $(r,\ell)$-Vertex Deletion problem, given a graph $G$ and an integer $k$, one has to decide whether at most $k$ vertices can be removed from $G$ to obtain an $(r,\ell)$-graph. This problem is NP-hard if $r+\ell \geq 1$ and encompasses several relevant problems such as Vertex Cover and Odd Cycle Transversal. The parameterized complexity of $(r,\ell)$-Vertex Deletion was known for all values of $(r,\ell)$ except for $(2,1)$, $(1,2)$, and $(2,2)$. We prove that each of these three cases is FPT and, furthermore, solvable in single-exponential time, which is asymptotically optimal in terms of $k$. We consider as well the version of $(r,\ell)$-Vertex Deletion where the set of vertices to be removed has to induce an independent set, and provide also a parameterized complexity dichotomy for this problem.