Multivariate Exploration of Metric Dilation

2025-01-08

AI Summary

This study investigates the Metric Inflation Augmentation problem: given a graph $G$, a target inflation factor $t$, and at most $k$ edges to add, decide whether augmenting $G$ yields a graph $Gamma$ whose shortest-path metric has inflation at most $t$. We establish the first fine-grained complexity dichotomy linking $t$ to structural graph parameters—maximum degree $Delta$, sparsity, and $K_{d,d}$-freeness—revealing a sharp computational phase transition between $t leq 2$ and $t geq 3$: for $t leq 2$, an FPT algorithm exists on sparse $K_{d,d}$-free graphs; for $t geq 3$, the problem is $W[1]$-hard even on star forests. We further devise an FPT algorithm parameterized by $k + t + Delta$, and precisely delineate the problem’s tractability boundary via $W[1]$- and $W[2]$-hardness reductions under structural constraints—including star forests and $K_{d,d}$-free graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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 $Gamma$. Even in this specific case, Dilation $t$-Augmentation remains computationally challenging. In particular, the problem is W[2]-hard parameterized by $k$ when $Gamma$ 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 $tleq 2$ and the problem is W[1]-hard for $tgeq 3$ even if $G$ is a forest consisting of disjoint stars. -- The problem is FPT parameterized by the combined parameter $k+t+Delta$, where $Delta$ is the maximum degree of the graph $G$ or $Gamma$.

Problem

• Graph Theory
• Geometric Embedding
• Algorithmic Complexity

Innovation

• Graph Expansion
• Parameterized Complexity
• Shortest Path