insight - Algorithms and Data Structures - # Metric Dimension and Geodetic Set Parameterized by Vertex Cover

Tight Bounds on Parameterized Complexity of Metric Dimension and Geodetic Set Problems by Vertex Cover

Q: Can the techniques developed in this paper be applied to other metric-based graph problems to obtain similar tight bounds

The techniques developed in this paper, such as the Set Identifying Gadget and Vertex Selector Gadgets, can potentially be applied to other metric-based graph problems to obtain similar tight bounds. By carefully constructing specialized gadgets that ensure certain properties in the graph, it is possible to create reductions that lead to lower bounds based on specific parameters. These techniques rely on clever graph constructions and the introduction of critical pairs and portals to control the resolution of vertices in the graph. Therefore, similar approaches could be explored for other metric-based graph problems to establish tight bounds based on different parameters.

Q: Are there any other structural parameters, besides vertex cover, for which tight bounds can be established for Metric Dimension and Geodetic Set

While vertex cover has been shown to be a structural parameter for which tight bounds can be established for Metric Dimension and Geodetic Set, there may be other structural parameters that could also lead to similar results. Parameters related to graph properties such as treewidth, feedback vertex set number, or modular-width have been studied in the context of other graph problems and could potentially be explored for Metric Dimension and Geodetic Set as well. By investigating how these structural parameters interact with the complexity of the problems, it may be possible to establish tight bounds and gain further insights into the algorithmic properties of Metric Dimension and Geodetic Set.

Q: What are the implications of these tight bounds on the practical applications of these problems, and how can they guide the design of more efficient algorithms

The tight bounds established for Metric Dimension and Geodetic Set based on the vertex cover parameter have significant implications for their practical applications and algorithm design. These bounds provide a clear understanding of the inherent complexity of these problems and help in identifying the limits of efficient algorithm design. By knowing that certain parameterized algorithms are unlikely to exist within certain time bounds, researchers and practitioners can focus on developing alternative approaches or heuristic methods to tackle these problems. Additionally, the insights gained from these bounds can guide the development of more efficient algorithms by highlighting the key structural properties that impact the complexity of Metric Dimension and Geodetic Set. This can lead to the design of specialized algorithms that leverage these properties to improve performance and scalability in real-world applications.

Core Concepts

The article establishes tight bounds on the parameterized complexity of the Metric Dimension and Geodetic Set problems with respect to the vertex cover number of the input graph. It provides FPT algorithms and matching lower bounds based on the Exponential Time Hypothesis.

Abstract

The paper studies the parameterized complexity of two metric-based graph problems, Metric Dimension and Geodetic Set, with respect to the vertex cover number (vc) of the input graph.

Key highlights:

The authors provide FPT algorithms for both problems running in time 2^O(vc^2) * n^O(1), and kernelization algorithms that output kernels with 2^O(vc) vertices.
They prove that, unless the Exponential Time Hypothesis (ETH) fails, these problems do not admit FPT algorithms running in time 2^o(vc^2) * n^O(1), nor kernelization algorithms that reduce the solution size and output kernels with 2^o(vc) vertices, even on graphs of bounded diameter.
These results constitute a rare set of tight lower bounds in parameterized complexity, with only a few other problems known to exhibit such tight bounds.
The authors develop novel techniques involving set identifying gadgets and vertex selector gadgets to establish the lower bounds.
The results provide a clear boundary between parameters yielding single-exponential and double-exponential running times for these two problems.

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Metric Dimension and Geodetic Set Parameterized by Vertex Cover

by Florent Fouc... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.01344.pdf

Metric Dimension and Geodetic Set Parameterized by Vertex Cover

Deeper Inquiries

Can the techniques developed in this paper be applied to other metric-based graph problems to obtain similar tight bounds

The techniques developed in this paper, such as the Set Identifying Gadget and Vertex Selector Gadgets, can potentially be applied to other metric-based graph problems to obtain similar tight bounds. By carefully constructing specialized gadgets that ensure certain properties in the graph, it is possible to create reductions that lead to lower bounds based on specific parameters. These techniques rely on clever graph constructions and the introduction of critical pairs and portals to control the resolution of vertices in the graph. Therefore, similar approaches could be explored for other metric-based graph problems to establish tight bounds based on different parameters.

Are there any other structural parameters, besides vertex cover, for which tight bounds can be established for Metric Dimension and Geodetic Set

While vertex cover has been shown to be a structural parameter for which tight bounds can be established for Metric Dimension and Geodetic Set, there may be other structural parameters that could also lead to similar results. Parameters related to graph properties such as treewidth, feedback vertex set number, or modular-width have been studied in the context of other graph problems and could potentially be explored for Metric Dimension and Geodetic Set as well. By investigating how these structural parameters interact with the complexity of the problems, it may be possible to establish tight bounds and gain further insights into the algorithmic properties of Metric Dimension and Geodetic Set.

What are the implications of these tight bounds on the practical applications of these problems, and how can they guide the design of more efficient algorithms

The tight bounds established for Metric Dimension and Geodetic Set based on the vertex cover parameter have significant implications for their practical applications and algorithm design. These bounds provide a clear understanding of the inherent complexity of these problems and help in identifying the limits of efficient algorithm design. By knowing that certain parameterized algorithms are unlikely to exist within certain time bounds, researchers and practitioners can focus on developing alternative approaches or heuristic methods to tackle these problems. Additionally, the insights gained from these bounds can guide the development of more efficient algorithms by highlighting the key structural properties that impact the complexity of Metric Dimension and Geodetic Set. This can lead to the design of specialized algorithms that leverage these properties to improve performance and scalability in real-world applications.