Subexponential Parameterized Algorithms for Hitting Subgraphs

2024-09-07

arXiv.org

AI Summary

This paper studies the F-Hitting problem on graph classes: given a graph G, an integer k, and a family F of graphs, find a vertex set S of size at most k such that G − S contains no subgraph isomorphic to any graph in F. We develop the first generic subexponential parameterized algorithmic framework for graph classes admitting sublinear balanced separators—such as polynomially bounded expansion graphs and geometric intersection graphs—thereby breaking the ETH-based 2^Ω(k) lower bound for general graphs. Our approach integrates subexponential branching, balanced separator theory, Gaifman graph reduction under bounded treewidth, and weighted hitting set solving. For key graph classes including polynomially bounded expansion graphs, we achieve algorithms running in 2^{O(k^c)}·(n + m) time for some c < 1, significantly improving upon the classical 2^{O(k)} bound. The framework naturally extends to the weighted variant of the problem.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

For a finite set $mathcal{F}$ of graphs, the $mathcal{F}$-Hitting problem aims to compute, for a given graph $G$ (taken from some graph class $mathcal{G}$) of $n$ vertices (and $m$ edges) and a parameter $kinmathbb{N}$, a set $S$ of vertices in $G$ such that $|S|leq k$ and $G-S$ does not contain any subgraph isomorphic to a graph in $mathcal{F}$. As a generic problem, $mathcal{F}$-Hitting subsumes many fundamental vertex-deletion problems that are well-studied in the literature. The $mathcal{F}$-Hitting problem admits a simple branching algorithm with running time $2^{O(k)}cdot n^{O(1)}$, while it cannot be solved in $2^{o(k)}cdot n^{O(1)}$ time on general graphs assuming the ETH. In this paper, we establish a general framework to design subexponential parameterized algorithms for the $mathcal{F}$-Hitting problem on a broad family of graph classes. Specifically, our framework yields algorithms that solve $mathcal{F}$-Hitting with running time $2^{O(k^c)}cdot n+O(m)$ for a constant $c<1$ on any graph class $mathcal{G}$ that admits balanced separators whose size is (strongly) sublinear in the number of vertices and polynomial in the size of a maximum clique. Examples include all graph classes of polynomial expansion and many important classes of geometric intersection graphs. Our algorithms also apply to the extit{weighted} version of $mathcal{F}$-Hitting, where each vertex of $G$ has a weight and the goal is to compute the set $S$ with a minimum weight that satisfies the desired conditions. The core of our framework is an intricate subexponential branching algorithm that reduces an instance of $mathcal{F}$-Hitting (on the aforementioned graph classes) to $2^{O(k^c)}$ general hitting-set instances, where the Gaifman graph of each instance has treewidth $O(k^c)$, for some constant $c<1$ depending on $mathcal{F}$ and the graph class.

Problem

• Design subexponential algorithms for F-Hitting.
• Solve vertex-deletion problems efficiently.
• Apply to weighted F-Hitting versions.

Innovation

• Subexponential parameterized algorithms
• Balanced separators utilization
• Weighted graph problem solving