Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parameterized Geometric Graph Modification with Disk Scaling

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, Meirav Zehavi

TL;DR

The technical contributions encompass a novel combination of linear programming, branching, and kernelization techniques, along with a fresh application of bidimensionality theory to analyze the area covered by disks, which may have broader applicability.

Abstract

The parameterized analysis of graph modification problems represents the most extensively studied area within Parameterized Complexity. Given a graph $G$ and an integer $k\in\mathbb{N}$ as input, the goal is to determine whether we can perform at most $k$ operations on $G$ to transform it into a graph belonging to a specified graph class $\mathcal{F}$. Typical operations are combinatorial and include vertex deletions and edge deletions, insertions, and contractions. However, in many real-world scenarios, when the input graph is constrained to be a geometric intersection graph, the modification of the graph is influenced by changes in the geometric properties of the underlying objects themselves, rather than by combinatorial modifications. It raises the question of whether vertex deletions or adjacency modifications are necessarily the most appropriate modification operations for studying modifications of geometric graphs. We propose the study of the disk intersection graph modification through the scaling of disks. This operation is typical in the realm of topology control but has not yet been explored in the context of Parameterized Complexity. We design parameterized algorithms and kernels for modifying to the most basic graph classes: edgeless, connected, and acyclic. Our technical contributions encompass a novel combination of linear programming, branching, and kernelization techniques, along with a fresh application of bidimensionality theory to analyze the area covered by disks, which may have broader applicability.

Parameterized Geometric Graph Modification with Disk Scaling

TL;DR

The technical contributions encompass a novel combination of linear programming, branching, and kernelization techniques, along with a fresh application of bidimensionality theory to analyze the area covered by disks, which may have broader applicability.

Abstract

The parameterized analysis of graph modification problems represents the most extensively studied area within Parameterized Complexity. Given a graph and an integer as input, the goal is to determine whether we can perform at most operations on to transform it into a graph belonging to a specified graph class . Typical operations are combinatorial and include vertex deletions and edge deletions, insertions, and contractions. However, in many real-world scenarios, when the input graph is constrained to be a geometric intersection graph, the modification of the graph is influenced by changes in the geometric properties of the underlying objects themselves, rather than by combinatorial modifications. It raises the question of whether vertex deletions or adjacency modifications are necessarily the most appropriate modification operations for studying modifications of geometric graphs. We propose the study of the disk intersection graph modification through the scaling of disks. This operation is typical in the realm of topology control but has not yet been explored in the context of Parameterized Complexity. We design parameterized algorithms and kernels for modifying to the most basic graph classes: edgeless, connected, and acyclic. Our technical contributions encompass a novel combination of linear programming, branching, and kernelization techniques, along with a fresh application of bidimensionality theory to analyze the area covered by disks, which may have broader applicability.

Paper Structure

This paper contains 44 sections, 51 theorems, 6 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Topology Control and Disk Scaling as a Modification Operation
  3. Problem Definitions
  4. Shrinking to Achieve Edgelessness and Acyclicity
  5. Shrinking/Expanding to Connectivity
  6. Overview of the Technical Results
  7. Subexponential FPT Algorithms
  8. Expanding to achieve Connectivity.
  9. Polynomial Kernels and Compressions and their Consequences
  10. (Min) $k$-Shrinking to Independence
  11. FPT Algorithm for Min $k$-Shrinking to Independence.
  12. (Min) $k$-Shrinking to Acyclicity
  13. Approximation Schemes for (Min) $k$-Shrinking to Independence
  14. Outline
  15. Preliminaries
  16. ...and 29 more sections

Key Result

Theorem 1

There exists an algorithm that solves an instance $\mathcal{I} = (P, k, \alpha)$ of $k$-Shrinking to Independence in time $2^{\mathcal{O}((\frac{1}{\alpha})^2\sqrt{k})}\cdot n^{\mathcal{O}(1)}$.

Figures (12)

  • Figure 1: Left: Original intersection graph $G = G(P, \bm{1})$ denoting an instance $(P, 1/2, 3)$ of $k$-Shrinking to Independence. Right: Edgeless graph $G(P, r)$ that shrinks three disks to radius $1/2$ (shown in red).
  • Figure 2: Left: Original intersection graph $G = G(P, \bm{1})$ corresponding to an instance $(P, 1/2, 3)$ of $k$-Shrinking to Acyclicity. Note that $\left\{ u, v\right\}$ is a minimum feedback vertex set for $G$. However, even after shrinking the corresponding disks to $1/2$ (shown in blue), the edges $\left\{ uw_1, uw_2, vx_1, v_2\right\}$ (shown in red) are present in the resulting intersection graph, showing that shrinking disks is not equivalent to vertex deletion. Right: A solution that shrinks three disks (shown in red) to radius $1/2$ corresponding to $u, x_2, y$, resulting in an acyclic graph $G(P, r)$.
  • Figure 3: Left: Original intersection graph $G = G(P, \bm{1})$ corresponding to an instance $(P, 1/2, 4)$ of $k$-Shrinking to Connectivity. Right: A solution that shrinks $4$ disks to radius $1/2$ while maintaining connectivity in the resulting intersection graph $G(P, r)$.
  • Figure 4: There is a solution that shrinks some of the disks in the clique; however, shrinking all the disks in the clique results in the disconnected graph (due to the red disks)
  • Figure 5: Each disk corresponding to the vertices of the grid minor is "touching" its neighbors, and shrinking any of the disks results in a disconnected graph.
  • ...and 7 more figures

Theorems & Definitions (94)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 84 more