Subexponential-time algorithms for finding large induced sparse subgraphs

TL;DR

This paper introduces subexponential algorithms for finding large induced subgraphs within certain hereditary graph classes, leveraging properties like sparsity, balanced separators, and fixed-parameter tractability.

Contribution

It establishes conditions under which large induced subgraphs can be found in subexponential time, extending algorithmic techniques to broader graph classes.

Findings

Largest induced forest in P_t-free graphs found in 2^{O(n^{2/3})} time

Largest induced planar graph in string graphs found in 2^{O(n^{3/4})} time

Provides a framework for subexponential algorithms based on graph properties

Abstract

Let $\mathcal{C}$ and $\mathcal{D}$ be hereditary graph classes. Consider the following problem: given a graph $G\in\mathcal{D}$, find a largest, in terms of the number of vertices, induced subgraph of $G$ that belongs to $\mathcal{C}$. We prove that it can be solved in $2^{o(n)}$ time, where $n$ is the number of vertices of $G$, if the following conditions are satisfied: * the graphs in $\mathcal{C}$ are sparse, i.e., they have linearly many edges in terms of the number of vertices; * the graphs in $\mathcal{D}$ admit balanced separators of size governed by their density, e.g., $\mathcal{O}(\Delta)$ or $\mathcal{O}(\sqrt{m})$, where $\Delta$ and $m$ denote the maximum degree and the number of edges, respectively; and * the considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph. This leads, for example, to the following corollaries for specific classes $\mathcal{C}$ and $\mathcal{D}$: * a largest induced forest in a $P_t$-free graph can be found in $2^{\tilde{\mathcal{O}}(n^{2/3})}$ time, for every fixed $t$; and * a largest induced planar graph in a string graph can be found in $2^{\tilde{\mathcal{O}}(n^{3/4})}$ time.