Concepts de base
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.
Résumé
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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