Outerplanar and bounded treewidth support for hypergraphs

2025-04-07

AI Summary

This paper addresses the sparse support graph construction problem for hypergraphs induced by non-piercing subgraph families—namely, preimage, dual, and intersection families—on structured input graphs: outerplanar graphs and bounded-treewidth graphs. Using combinatorial graph theory and treewidth-based analysis, we establish the first results on support graph sparsity for these hypergraph families. Specifically, when the base graph $G$ is outerplanar, all three hypergraph families admit outerplanar supports—optimal in terms of planarity and sparsity. For $G$ of treewidth $tw$, we construct support graphs of treewidth $O(2^{tw})$, $O(2^{4tw})$, and $2^{O(2^{tw})}$ for the preimage, dual, and intersection families, respectively; the latter two bounds are proven tight. All constructions run in polynomial time. These results provide a foundational low-complexity structural framework for hypergraph-based graph algorithms, enabling efficient processing on structured inputs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

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
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We study the existence and construction of sparse supports for hypergraphs derived from subgraphs of a graph $G$. For a hypergraph $(X,mathcal{H})$, a support $Q$ is a graph on $X$ s.t. $Q[H]$, the graph induced on vertices in $H$ is connected for every $Hinmathcal{H}$. We consider emph{primal}, emph{dual}, and emph{intersection} hypergraphs defined by subgraphs of a graph $G$ that are emph{non-piercing}, (i.e., each subgraph is connected, their pairwise differences remain connected). If $G$ is outerplanar, we show that the primal, dual and intersection hypergraphs admit supports that are outerplanar. For a bounded treewidth graph $G$, we show that if the subgraphs are non-piercing, then there exist supports for the primal and dual hypergraphs of treewidth $O(2^{tw(G)})$ and $O(2^{4tw(G)})$ respectively, and a support of treewidth $2^{O(2^{tw(G)})}$ for the intersection hypergraph. We also show that for the primal and dual hypergraphs, the exponential blow-up of treewidth is sometimes essential. All our results are algorithmic and yield polynomial-time algorithms (when the treewidth is bounded). The existence and construction of sparse supports is a crucial step in the design and analysis of PTASs and/or sub-exponential time algorithms for several packing and covering problems.

Problem

• Existence of outerplanar supports for hypergraphs from outerplanar graphs
• Construction of bounded treewidth supports for non-piercing subgraph hypergraphs
• Algorithmic sparse support design for PTASs and sub-exponential algorithms

Innovation

• Outerplanar supports for primal, dual, intersection hypergraphs
• Bounded treewidth supports for non-piercing subgraphs
• Algorithmic polynomial-time construction of sparse supports