Polynomial-time approximation schemes for induced subgraph problems on fractionally tree-independence-number-fragile graphs

TL;DR

The paper introduces efficient fractional tree-alpha-fragility and layered tree-independence number as unifying frameworks to obtain PTASes for Max Weight Induced Subgraph and Max Weight Independent Packing on broad graph classes, including intersection graphs of fat objects in fixed dimensions and minor-closed families. By proving that such classes are efficiently fractionally tree-alpha-fragile and bounded in layered tree-independence number, it extends existing PTAS meta-theorems to a wider range of problems expressible in $\mathsf{CMSO}_2$. It also derives subexponential-time algorithms for distance-$d$ packing on these classes and analyzes the closure properties under odd powers, illustrating both the reach and limitations of the approach (e.g., non-existence of EPTAS under standard complexity assumptions). Overall, the work provides a cohesive, geometric-graph framework that explains and extends many PTAS results for induced-subgraph problems, with broad implications for both theory and applicability in geometric intersection graphs.

Abstract

We investigate a relaxation of the notion of fractional treewidth-fragility, namely fractional tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for meta-problems such as finding a maximum-weight sparse induced subgraph satisfying a given $\mathsf{CMSO}_2$ formula on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth, and its applications to exact subexponential-time algorithms.

Figures (3)

• Figure 1: Relationships between the main graph classes related to the paper, where an arrow represents class inclusion or implication between class properties. The following shorthands are adopted. $\mathsf{tw}$ and $\mathsf{tree}\textnormal{-}\alpha$ are shorthands for treewidth and tree-independence-number, respectively. bounded number of crossings per edge stands for the class of graphs embeddable on a surface of bounded genus with a bounded number of crossings per edge. unit disks, disks, and fat objects are shorthands for the class of intersection graphs of a collection of unit disks in the plane, disks in the plane, and $c$-fat objects in some $d$-dimensional space, respectively. We reference only the inclusions or implications not directly following from the definition. A dashed red arrow indicates that determining whether the corresponding inclusion holds is, to the best of our knowledge, open.
• Figure 2: A comb $O_n$ with $(n+1)/2$ teeth each of width $1/n$, for some odd $n \in \mathbb{N}$. It is easy to see that $O_n$ is $\sqrt{\pi}$-thick but the collection $\{O_n : n \in \mathbb{N}\}$ is not $k$-globally fat, for any $k \geq 1$.
• Figure 3: Realizing an outerstring graph as the intersection graph of a globally fat collection of objects in $\mathbb{R}^2$.

Theorems & Definitions (11)

• Remark 1
• Remark 2
• Definition 3
• Remark 4
• Lemma 5: Folklore
• Lemma 6
• Lemma 7
• Corollary 8
• Lemma 9
• Lemma 10