# Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms

## TL;DR

This paper introduces two combinatorial tools: a randomized algorithm for finding large independent sets in d-degenerate graphs and a graph sparsification method that preserves minimal multicuts, enabling new fixed parameter tractable algorithms.

## Contribution

The paper presents novel randomized and deterministic tools that improve fixed parameter algorithms for several graph problems, resolving open questions.

## Key findings

- Developed a linear-time randomized algorithm for independent sets in d-degenerate graphs.
- Created a polynomial-time graph sparsification procedure preserving minimal multicuts.
- Enabled new FPT algorithms for Stable s-t Separator, Stable Odd Cycle Transversal, and other problems.

## Abstract

We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a $d$-degenerate graph $G$ and an integer $k$, outputs an independent set $Y$, such that for every independent set $X$ in $G$ of size at most $k$, the probability that $X$ is a subset of $Y$ is at least $\left({(d+1)k \choose k} \cdot k(d+1)\right)^{-1}$.The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph $G$, a set $T = \{\{s_1, t_1\}, \{s_2, t_2\}, \ldots, \{s_\ell, t_\ell\}\}$ of terminal pairs and an integer $k$, returns an induced subgraph $G^\star$ of $G$ that maintains all the inclusion minimal multicuts of $G$ of size at most $k$, and does not contain any $(k+2)$-vertex connected set of size $2^{{\cal O}(k)}$. In particular, $G^\star$ excludes a clique of size $2^{{\cal O}(k)}$ as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for Stable $s$-$t$ Separator, Stable Odd Cycle Transversal and Stable Multicut on general graphs, and for Stable Directed Feedback Vertex Set on $d$-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our algorithms can be derandomized at the cost of a small overhead in the running time.