Partitioning a Graph into Small Pieces with Applications to Path Transversal

TL;DR

This paper develops approximation algorithms for graph partitioning problems, including vertex and edge separators and path transversals, with improved ratios and running times, especially for small or slowly growing parameters.

Contribution

It introduces $O( ext{log }k)$-approximation algorithms for $k$-Vertex and $k$-Edge Separators, and for $k$-Path Transversal, with novel analysis and faster algorithms for small $k$.

Findings

Achieved $O( ext{log }k)$-approximation for $k$-Vertex Separator with $2^{O(k)} n^{O(1)}$ time.

Achieved $O( ext{log }k)$-approximation for $k$-Edge Separator with polynomial time.

Provided an $O( ext{log }k)$-approximation for $k$-Path Transversal with $2^{O(k^3 ext{log }k)} n^{O(1)}$ time.

Abstract

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 $k^{\Omega(OPT)} n^{\Omega(1)}$ 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 $(1 - \delta)k$-approximation algorithm for fixed $\delta > 0$ was open.