Metric Dimension and Geodetic Set Parameterized by Vertex Cover

2024-05-02

Symposium on Theoretical Aspects of Computer Science

AI Summary

This paper investigates the parameterized complexity of Metric Dimension and Geodetic Set on graphs, parameterized by vertex cover number $vc$. Both problems are NP-hard, and we establish the first tight conditional lower bounds: assuming the Exponential Time Hypothesis (ETH), neither problem admits a $2^{o(vc^2)}$-time algorithm nor a $2^{o(vc)}$-vertex kernel. Complementing these, we provide matching upper bounds via $2^{O(vc^2)}n^{O(1)}$-time FPT algorithms and $2^{O(vc)}$-vertex kernels. Our techniques integrate ETH-based reductions, structural analysis of distance properties, and novel kernelization design, and extend to graphs of bounded diameter. This work precisely characterizes the optimal FPT complexity of both problems with respect to $vc$, establishing their FPT optimality under ETH. Moreover, the methodological framework is generalizable across related graph problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data

Abstract

For a graph $G$, a subset $Ssubseteq V(G)$ is called a resolving set of $G$ if, for any two vertices $u,vin V(G)$, there exists a vertex $win S$ such that $d(w,u) eq d(w,v)$. The Metric Dimension problem takes as input a graph $G$ on $n$ vertices and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. In another metric-based graph problem, Geodetic Set, the input is a graph $G$ and an integer $k$, and the objective is to determine whether there exists a subset $Ssubseteq V(G)$ of size at most $k$ such that, for any vertex $u in V(G)$, there are two vertices $s_1, s_2 in S$ such that $u$ lies on a shortest path from $s_1$ to $s_2$. These two classical problems turn out to be intractable with respect to the natural parameter, i.e., the solution size, as well as most structural parameters, including the feedback vertex set number and pathwidth. Some of the very few existing tractable results state that they are both FPT with respect to the vertex cover number $vc$. More precisely, we observe that both problems admit an FPT algorithm running in time $2^{mathcal{O}(vc^2)}cdot n^{mathcal{O}(1)}$, and a kernelization algorithm that outputs a kernel with $2^{mathcal{O}(vc)}$ vertices. We prove that unless the Exponential Time Hypothesis fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, neither admit an FPT algorithm running in time $2^{o(vc^2)}cdot n^{mathcal(1)}$, nor a kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(vc)}$ vertices. The versatility of our technique enables us to apply it to both these problems. We only know of one other problem in the literature that admits such a tight lower bound. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

Problem

• Solves Metric Dimension problem for graphs using resolving sets.
• Addresses Geodetic Set problem via shortest path vertex subsets.
• Proves FPT algorithms and kernelization bounds for vertex cover parameter.

Innovation

• FPT algorithm with vertex cover parameterization
• Kernelization algorithm for graph problems
• Tight lower bounds under Exponential Time Hypothesis