Core Concepts

The locality number of a word is closely related to the cutwidth and pathwidth of graphs, allowing for efficient algorithms to compute or approximate the locality number.

Abstract

The paper investigates the computational complexity of the locality number, a recently introduced structural parameter for strings, and its connections to the well-studied graph parameters of cutwidth and pathwidth.
Key highlights:
The authors establish approximation-preserving reductions between the problems of computing the locality number, cutwidth, and pathwidth. This allows them to show that:
Computing the locality number is NP-hard, but fixed-parameter tractable when parameterized by the locality number or the alphabet size.
There is a polynomial-time O(√log(opt) log(n))-approximation algorithm for the minimization version of the locality number problem.
The authors identify direct connections between the string parameter of the locality number and the graph parameters of cutwidth and pathwidth. These connections lead to new approximation algorithms for cutwidth:
There is a polynomial-time O(√log(opt) log(h))-approximation algorithm for the minimization version of cutwidth on multigraphs with h edges.
There is a polynomial-time O(√log(opt) opt)-approximation algorithm for the minimization version of cutwidth on multigraphs.
The authors investigate the performance of natural greedy strategies for approximating the locality number and show that they do not yield good approximation algorithms.
The authors provide a direct approximation-preserving reduction from cutwidth to pathwidth, which allows them to translate known approximation results for pathwidth into new approximation results for cutwidth.

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Key Insights Distilled From

by Katrin Casel... at **arxiv.org** 04-26-2024

Deeper Inquiries

One other string parameter that could be related to graph parameters in a similar way is the "treewidth" of a string. Treewidth is a graph parameter that measures how tree-like a graph is, and it has connections to various algorithmic problems. By defining a suitable transformation from a string to a graph representation that captures the treewidth of the string, we can establish a relationship between the treewidth of the string and graph parameters like cutwidth and pathwidth. This connection can be exploited algorithmically by leveraging known algorithms and results for treewidth in graph theory to solve problems related to cutwidth and pathwidth of the corresponding graph representations of strings.

The approximation ratios for cutwidth can potentially be improved by delving deeper into the connection to the locality number. By exploring more intricate relationships between the structural properties of words that lead to high locality numbers and the corresponding graph parameters, new insights may be gained. This deeper understanding could lead to the development of more sophisticated approximation algorithms that take advantage of the specific characteristics of words with high locality numbers. Additionally, refining the reduction from cutwidth to pathwidth via the locality number could potentially uncover new approximation techniques that yield better results for cutwidth approximation.

The structural properties of words that result in high locality numbers are closely related to concepts in combinatorics on words such as unavoidable patterns and word equations. Unavoidable patterns are patterns that must occur in every infinite word over a given alphabet, and they are often characterized by specific repetitive structures. Similarly, words with high locality numbers exhibit repetitive and alternating patterns that contribute to their complexity. Understanding the connections between these structural properties can provide insights into the inherent complexity of pattern matching and word equations, shedding light on the fundamental properties of strings and their algorithmic behavior. By studying these relationships, researchers can uncover deeper connections between different areas of combinatorics on words and develop more efficient algorithms for solving related problems.

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