Multivariate Exploration of Metric Dilation
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arXiv
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2025
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| author | Banik, Aritra Fomin, Fedor V. Golovach, Petr A. Inamdar, Tanmay Jana, Satyabrata Saurabh, Saket |
| author_facet | Banik, Aritra Fomin, Fedor V. Golovach, Petr A. Inamdar, Tanmay Jana, Satyabrata Saurabh, Saket |
| contents | 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 $Γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04555 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multivariate Exploration of Metric Dilation Banik, Aritra Fomin, Fedor V. Golovach, Petr A. Inamdar, Tanmay Jana, Satyabrata Saurabh, Saket Discrete Mathematics Computational Geometry Combinatorics 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 $Γ$. |
| title | Multivariate Exploration of Metric Dilation |
| topic | Discrete Mathematics Computational Geometry Combinatorics |
| url | https://arxiv.org/abs/2501.04555 |