On the Parameterized Complexity Of Grid Contraction

TL;DR

This paper studies the computational complexity of the Grid Contraction problem, providing fixed parameter tractable algorithms and polynomial kernels, and establishing lower bounds under ETH.

Contribution

It introduces the parameterized complexity analysis of Grid Contraction, offering new algorithms and kernelization results, and proving complexity lower bounds.

Findings

Fixed parameter tractable algorithm with runtime c^k * polynomial

Polynomial kernel for Grid Contraction

Conditional lower bounds based on ETH

Abstract

For a family of graphs $\mathcal{G}$, the $\mathcal{G}$-\textsc{Contraction} problem takes as an input a graph $G$ and an integer $k$, and the goal is to decide if there exists $F \subseteq E(G)$ of size at most $k$ such that $G/F$ belongs to $\mathcal{G}$. Here, $G/F$ is the graph obtained from $G$ by contracting all the edges in $F$. In this article, we initiate the study of \textsc{Grid Contraction} from the parameterized complexity point of view. We present a fixed parameter tractable algorithm, running in time $c^k \cdot |V(G)|^{\mathcal{O}(1)}$, for this problem. We complement this result by proving that unless \ETH\ fails, there is no algorithm for \textsc{Grid Contraction} with running time $c^{o(k)} \cdot |V(G)|^{\mathcal{O}(1)}$. We also present a polynomial kernel for this problem.