Finding large sparse induced subgraphs in graphs of small (but not very small) tree-independence number

2026-01-22

AI Summary

This work addresses the problem of efficiently computing a maximum-weight induced subgraph satisfying a CMSO₂-definable property in graphs whose tree-independence number is small (e.g., polylogarithmic). By integrating tree decompositions, CMSO₂ logical expressiveness, balanced clique separators, and parameterized algorithmic techniques, the authors devise a novel algorithm with running time \(n^{O(k)}\), where \(k\) denotes the tree-independence number. This approach substantially improves upon previous complexity bounds, yielding the first quasipolynomial-time algorithm for super-constant values of \(k\). Furthermore, when applied to geometric intersection graphs admitting sublinear balanced clique separators, the method achieves subexponential time performance, thereby significantly broadening the class of graph optimization problems that can be solved efficiently under structural sparsity conditions.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The independence number of a tree decomposition is the size of a largest independent set contained in a single bag. The tree-independence number of a graph $G$ is the minimum independence number of a tree decomposition of $G$. As shown recently by Lima et al. [ESA~2024], a large family of optimization problems asking for a maximum-weight induced subgraph of bounded treewidth, satisfying a given \textsf{CMSO}$_2$ property, can be solved in polynomial time in graphs whose tree-independence number is bounded by some constant~$k$. However, the complexity of the algorithm of Lima et al. grows rapidly with $k$, making it useless if the tree-independence number is superconstant. In this paper we present a refined version of the algorithm. We show that the same family of problems can be solved in time~$n^{\mathcal{O}(k)}$, where $n$ is the number of vertices of the instance, $k$ is the tree-independence number, and the $\mathcal{O}(\cdot)$-notation hides factors depending on the treewidth bound of the solution and the considered \textsf{CMSO}$_2$ property. This running time is quasipolynomial for classes of graphs with polylogarithmic tree-independence number; several such classes were recently discovered. Furthermore, the running time is subexponential for many natural classes of geometric intersection graphs -- namely, ones that admit balanced clique-based separators of sublinear size.

Problem

• tree-independence number
• induced subgraph
• sparse graphs
• algorithmic complexity
• CMSO_2 properties

Innovation

• tree-independence number
• induced subgraph
• CMSO₂ logic
• quasipolynomial time
• geometric intersection graphs