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Polynomial-Time Approximation Schemes for Independent Packing Problems on Fractionally Tree-Independence-Number-Fragile Graphs

Esther Galby, Andrea Munaro, Shizhou Yang

TL;DR

This work advances approximation algorithms for independent packing by introducing and leveraging fractional tree-alpha-fragility through the lens of tree-independence and layered structures. By defining and exploiting layered tree-independence numbers, the authors unify PTAS results across geometric intersection graphs (such as fat objects and unit disks) and minor-closed classes, and extend these techniques to distance-d variants. A general cover-based PTAS framework is developed, producing efficient algorithms for Max Weight Independent Packing on efficiently fractionally tree-alpha-fragile classes and its geometric specializations, while clarifying limits such as the absence of an EPTAS under standard complexity assumptions. The results have practical impact for a broad class of geometric and graph- theoretic problems, enabling scalable PTASes for packing problems in fat-object intersections and grid-structured domains.

Abstract

We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems 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.

Polynomial-Time Approximation Schemes for Independent Packing Problems on Fractionally Tree-Independence-Number-Fragile Graphs

TL;DR

This work advances approximation algorithms for independent packing by introducing and leveraging fractional tree-alpha-fragility through the lens of tree-independence and layered structures. By defining and exploiting layered tree-independence numbers, the authors unify PTAS results across geometric intersection graphs (such as fat objects and unit disks) and minor-closed classes, and extend these techniques to distance-d variants. A general cover-based PTAS framework is developed, producing efficient algorithms for Max Weight Independent Packing on efficiently fractionally tree-alpha-fragile classes and its geometric specializations, while clarifying limits such as the absence of an EPTAS under standard complexity assumptions. The results have practical impact for a broad class of geometric and graph- theoretic problems, enabling scalable PTASes for packing problems in fat-object intersections and grid-structured domains.

Abstract

We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems 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.
Paper Structure (12 sections, 29 theorems, 1 figure)

This paper contains 12 sections, 29 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Our results
  3. Other related work
  4. Preliminaries
  5. Layered and local tree-independence number
  6. Intersection graphs of bounded layered tree-independence number
  7. Fractional $\mathrm{tree}\textnormal{-}\alpha$-fragility
  8. Intersection graphs of fat objects
  9. PTASes
  10. Packing subgraphs at distance at least $1$ in efficiently fractionally $\mathrm{tree}\textnormal{-}\alpha$-fragile classes
  11. Packing subgraphs at distance at least $d$ in graphs with bounded layered tree-independence number
  12. Packing independent unit disks, unit-width rectangles and paths with bounded horizontal part on a grid

Key Result

Theorem 1

Max Weight Independent Packing admits a PTAS on every efficiently fractionally $\mathrm{tree}\textnormal{-}\alpha$-fragile class. Moreover, the class of intersection graphs of fat objects in $\mathbb{R}^d$, for fixed $d$, is efficiently fractionally $\mathrm{tree}\textnormal{-}\alpha$-fragileHere we

Figures (1)

  • Figure 1: Examples showing that VPG/EPG graphs and intersection graphs of rectangles have unbounded layered tree-independence number: VPG/EPG representation (left) and representation by intersection of rectangles (right) of $K_{4,4}$.

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Corollary 6
  • Theorem 7
  • Corollary 8
  • Lemma 9: Milanič and Rza̧żewski MR22
  • Theorem 10
  • ...and 23 more