A Single Exponential-Time FPT Algorithm for Cactus Contraction
R. Krithika, Pranabendu Misra, Prafullkumar Tale
TL;DR
This work studies Cactus Contraction, the problem of contracting at most $k$ edges to obtain a cactus. It introduces a framework based on a $T$-witness structure and compatible colorings, then employs a recoloring refinement to isolate big witness sets and extract connected cores for a safe contraction. The authors present a randomized $2^{\mathcal{O}(k)} \cdot |V(G)|^{\mathcal{O}(1)}$-time algorithm for 2-connected graphs and extend it to general graphs, followed by a derandomization via universal sets to obtain a deterministic $2^{\mathcal{O}(k)} \cdot |V(G)|^{\mathcal{O}(1)}$-time algorithm. The result contributes a first single-exponential-time FPT algorithm for cactus contraction, advancing our understanding of contraction problems beyond trees and paths and enabling potential extensions to other bounded-treewidth targets.
Abstract
For a collection $\mathcal{F}$ of graphs, the $\mathcal{F}$-\textsc{Contraction} problem takes a graph $G$ and an integer $k$ as input and decides if $G$ can be modified to some graph in $\mathcal{F}$ using at most $k$ edge contractions. The $\mathcal{F}$-\textsc{Contraction} problem is \NP-Complete for several graph classes $\mathcal{F}$. Heggerners et al. [Algorithmica, 2014] initiated the study of $\mathcal{F}$-\textsc{Contraction} in the realm of parameterized complexity. They showed that it is \FPT\ if $\mathcal{F}$ is the set of all trees or the set of all paths. In this paper, we study $\mathcal{F}$-\textsc{Contraction} where $\mathcal{F}$ is the set of all cactus graphs and show that we can solve it in $2^{\calO(k)} \cdot |V(G)|^{\OO(1)}$ time.