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

TL;DR

The paper addresses maximizing the weight of an induced subgraph with bounded treewidth that satisfies a CMSO$_2$ property, in graphs whose tree-independence number is bounded by $k$. It refines a prior meta-theorem to achieve a running time of $n^{\mathcal{O}(k)}$, with constants depending on the fixed $t$ and the CMSO$_2$ sentence, by introducing boundaried-graph types and a signature-based state compression within a bottom-up dynamic program over a nice tree decomposition. This yields quasipolynomial-time results for graph classes with polylogarithmic $k$ (e.g., certain 3PC- and hole-free families) and subexponential-time results for geometric intersection graphs with sublinear balanced clique-based separators, such as convex fat objects in $\mathbb{R}^2$, map graphs, and pseudodisk graphs. The approach leverages CMSO$_2$ expressibility, boundaried-graph composition, and a carefully defined signature framework to bound the state space, enabling efficient computation of a maximum-weight feasible subgraph. Overall, the work broadens the applicability of CMSO-based meta-theorems to a wider set of graph families by achieving a tighter, single-exponential dependence on the tree-independence parameter and providing practical consequences for dense and geometric graph classes.

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.

Theorems & Definitions (23)

• Theorem 1.1: Lima et al. DBLP:conf/esa/LimaMMORS24
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Chudnovsky et al. DBLP:journals/jctb/ChudnovskyHLS26, Chudnovsky et al. DBLP:conf/soda/ChudnovskyGHLS25, Chudnovsky et al. DBLP:journals/corr/abs-2501-14658, Chudnovsky et al. DBLP:journals/corr/abs-2509-15458
• Corollary 1.5
• Theorem 1.6: de Berg et al. DBLP:journals/siamcomp/BergBKMZ20, de Berg et al. DBLP:journals/algorithmica/BergKMT23, Dallard et al. DBLP:journals/corr/abs-2506-12424
• Lemma 1.7
• Theorem 1.8
• Theorem 2.1: Bodlaender DBLP:journals/siamcomp/Bodlaender96
• Lemma 2.2