Core Concepts

Metric-based graph problems in NP, such as Metric Dimension, Strong Metric Dimension, and Geodetic Set, admit double-exponential lower bounds in the treewidth or vertex cover number of the input graph, unless the Exponential Time Hypothesis fails.

Abstract

The paper presents novel techniques to obtain double-exponential lower bounds in the treewidth (tw) or vertex cover number (vc) for three natural and well-studied NP-complete graph problems: Metric Dimension, Strong Metric Dimension, and Geodetic Set. These are the first problems in NP known to admit such lower bounds.
The key insights are:
The authors develop a reduction from the 3-Partitioned-3-SAT problem, which is known to require double-exponential time under the Exponential Time Hypothesis (ETH). The reduction encodes the relationships between clause and literal vertices using "small" separators, rather than the typical incidence graph representation.
For Metric Dimension and Geodetic Set, the authors prove that these problems do not admit algorithms running in time 2^(f(diam)^o(tw)) * n^O(1), for any computable function f, unless the ETH fails. This implies that these problems on graphs of bounded diameter cannot admit 2^(2^o(tw)) * n^O(1)-time algorithms, unless the ETH fails.
For Strong Metric Dimension, the authors show that the problem does not admit an algorithm running in time 2^(2^o(vc)) * n^O(1), unless the ETH fails. This also implies that Strong Metric Dimension does not admit a kernelization algorithm that outputs an instance with 2^o(vc) vertices, unless the ETH fails.
The authors complement their lower bounds with matching (and sometimes non-trivial) upper bounds, demonstrating the tightness of their results.
The authors believe that their novel technique based on Sperner families of sets will lead to obtaining similar double-exponential lower bounds for many other problems in NP.

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Key Insights Distilled From

by Florent Fouc... at **arxiv.org** 05-01-2024

Deeper Inquiries

The authors' technique for obtaining double-exponential lower bounds can potentially be extended to other natural NP-complete problems beyond Metric Dimension, Strong Metric Dimension, and Geodetic Set. The key lies in the versatility and simplicity of the technique based on Sperner families of sets. By encoding specific relationships between clause and literal vertices into small separators, it becomes feasible to apply this technique to a wide range of NP-complete graph problems.
The success of the technique in proving double-exponential lower bounds for the three considered problems showcases its potential for application to other NP-complete problems. As long as the problem involves intricate relationships between different parts of the input structure and can be encoded effectively using the proposed method, it is plausible to extend the technique to derive double-exponential lower bounds for various other NP-complete problems in graph theory and beyond.

In addition to treewidth and vertex cover, there are several other structural parameters for which the authors' technique can be applied to derive double-exponential lower bounds. Some potential structural parameters include pathwidth, branchwidth, tree-depth, and cliquewidth. These parameters are commonly used in parameterized complexity theory to analyze the computational complexity of problems on graphs.
The technique's effectiveness in proving double-exponential lower bounds for Metric Dimension, Strong Metric Dimension, and Geodetic Set parameterized by treewidth and vertex cover demonstrates its adaptability to different structural parameters. By appropriately modifying the construction and encoding of relationships between vertices based on the specific parameter, it is feasible to extend the technique to derive double-exponential lower bounds for problems parameterized by these alternative structural parameters.

The double-exponential lower bounds obtained using the authors' technique have significant applications and implications in the context of network design and monitoring problems. These lower bounds provide insights into the inherent computational complexity of solving Metric Dimension, Strong Metric Dimension, and Geodetic Set on graphs with bounded diameter or treewidth.
In network design, the lower bounds highlight the challenges in efficiently solving metric-based graph problems, even on restricted graph classes. They emphasize the computational hardness of finding optimal solutions for these problems, especially when considering the interplay between structural parameters like treewidth and vertex cover.
Furthermore, in network monitoring, the double-exponential lower bounds shed light on the limitations of algorithmic approaches for resolving Metric Dimension, Strong Metric Dimension, and Geodetic Set. These bounds serve as a benchmark for evaluating the efficiency of algorithms and provide guidance for developing improved computational methods for network monitoring applications.

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