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Multivariate Exploration of Metric Dilation

Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, Saket Saurabh

Abstract

Let $G$ be a weighted graph embedded in a metric space $(M, d_M )$. The vertices of $G$ correspond to the points in $M$ , with the weight of each edge $uv$ being the distance $d_M (u, v)$ between their respective points in $M$ . The dilation (or stretch) of $G$ is defined as the minimum factor $t$ such that, for any pair of vertices $u, v$, the distance between $u$ and $v$-represented by the weight of a shortest $u$, $v$-path is at most $ t \cdot d_M (u, v)$. We study Dilation t-Augmentation, where the objective is, given a metric $M $, a graph $G$, and numerical values $k$ and $t$, to determine whether $G$ can be transformed into a graph with dilation $t$ by adding at most $k$ edges. Our primary focus is on the scenario where the metric $M$ is the shortest path metric of an unweighted graph $Γ$. Even in this specific case, Dilation $t$-Augmentation remains computationally challenging. In particular, the problem is W[2]-hard parameterized by $k$ when $Γ$ is a complete graph, already for $t=2$. Our main contribution lies in providing new insights into the impact of combinations of various parameters on the computational complexity of the problem. We establish the following. -- The parameterized dichotomy of the problem with respect to dilation $t$, when the graph $G$ is sparse: Parameterized by $k$, the problem is FPT for graphs excluding a biclique $K_{d,d}$ as a subgraph for $t\leq 2$ and the problem is W[1]-hard for $t\geq 3$ even if $G$ is a forest consisting of disjoint stars. -- The problem is FPT parameterized by the combined parameter $k+t+Δ$, where $Δ$ is the maximum degree of the graph $G$ or $Γ$.

Multivariate Exploration of Metric Dilation

Abstract

Let be a weighted graph embedded in a metric space . The vertices of correspond to the points in , with the weight of each edge being the distance between their respective points in . The dilation (or stretch) of is defined as the minimum factor such that, for any pair of vertices , the distance between and -represented by the weight of a shortest , -path is at most . We study Dilation t-Augmentation, where the objective is, given a metric , a graph , and numerical values and , to determine whether can be transformed into a graph with dilation by adding at most edges. Our primary focus is on the scenario where the metric is the shortest path metric of an unweighted graph . Even in this specific case, Dilation -Augmentation remains computationally challenging. In particular, the problem is W[2]-hard parameterized by when is a complete graph, already for . Our main contribution lies in providing new insights into the impact of combinations of various parameters on the computational complexity of the problem. We establish the following. -- The parameterized dichotomy of the problem with respect to dilation , when the graph is sparse: Parameterized by , the problem is FPT for graphs excluding a biclique as a subgraph for and the problem is W[1]-hard for even if is a forest consisting of disjoint stars. -- The problem is FPT parameterized by the combined parameter , where is the maximum degree of the graph or .
Paper Structure (13 sections, 20 theorems, 2 equations, 3 figures, 1 table)

This paper contains 13 sections, 20 theorems, 2 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Model, Results, and Methods
  3. Notations.
  4. Conflict versus Adjacent Conflicts
  5. Dilation $2$-Augmentation for $\mathcal{K}_{d,d}$-free Graphs
  6. Conflict Graph and Vertex Cover
  7. Bounding the Size of the $V_c$ in Annotated Instances
  8. Solving Annotated Instances with bounded $V_c$
  9. FPT Algorithms of Dilation $t$-Augmentation for Bounded Degree Graphs
  10. $\Gamma$ is of Bounded Degree
  11. $G$ is of Bounded Degree
  12. Dilation Augmentation For Forest and Stars
  13. W[1]-hardness of Dilation $3$-Augmentation when $G$ is Forest

Key Result

Lemma 1

For any set of edges $S$, $G+S$ is conflict-free if and only if $G+S$ is adjacent conflict-free.

Figures (3)

  • Figure 1: An instance of Dilation $2$-Augmentation with $k=2$. The edges of the solution $S$ are shown in dashed red. The edge weights in $G$ are derived from the corresponding shortest path in $\Gamma$.
  • Figure 3: Path in $G+S$ is shown in red (solid lines), path in $\Gamma$ is shown in green (dashed) and path in $G$ is shown in blue ( dashed-dotted).
  • Figure 4: Hardness for $t=3$ when $G$ is a disjoint union of a star, and a set of isolated vertices.

Theorems & Definitions (21)

  • Lemma 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Definition 6: BonnetBTW19
  • Proposition 7
  • Lemma 8
  • Lemma 9
  • Lemma 11
  • Lemma 15
  • ...and 11 more