Partitioning a Graph into Small Pieces with Applications to Path Transversal

Given a graph $G = (V, E)$ and an integer $k$, we study $k$-Vertex Seperator (resp. $k$-Edge Separator), where the goal is to remove the minimum number of vertices (resp. edges) such that each connected component in the resulting graph has at most $k$ vertices. Our primary focus is on the case where $k$ is either a constant or a slowly growing function of $n$ (e.g. $O(\log n)$ or $n^{o(1)}$). Our problems can be interpreted as a special case of three general classes of problems that have been studied separately (balanced graph partitioning, Hypergraph Vertex Cover (HVC), and fixed parameter tractability (FPT)). Our main result is an $O(\log k)$-approximation algorithm for $k$-Vertex Seperator that runs in time $2^{O(k)} n^{O(1)}$, and an $O(\log k)$-approximation algorithm for $k$-Edge Separator that runs in time $n^{O(1)}$. Our result on $k$-Edge Seperator improves the best previous graph partitioning algorithm for small $k$. Our result on $k$-Vertex Seperator improves the simple $(k+1)$-approximation from HVC. When $OPT > k$, the running time $2^{O(k)} n^{O(1)}$ is faster than the lower bound $k^{Ω(OPT)} n^{Ω(1)}$ for exact algorithms assuming the Exponential Time Hypothesis. While the running time of $2^{O(k)} n^{O(1)}$ for $k$-Vertex Separator seems unsatisfactory, we show that the superpolynomial dependence on $k$ may be needed to achieve a polylogarithmic approximation ratio, based on hardness of Densest $k$-Subgraph. We also study $k$-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length $k$. With additional ideas from FPT algorithms and graph theory, we present an $O(\log k)$-approximation algorithm for $k$-Path Transversal that runs in time $2^{O(k^3 \log k)} n^{O(1)}$. Previously, the existence of even $(1 - δ)k$-approximation algorithm for fixed $δ> 0$ was open.