We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard, but fixed-parameter tractable, if parameterised by the locality number or by the alphabet size, which has been formulated as open problems in the literature. Moreover, the locality number can be approximated with ratio O(sqrt(log(opt)) log(n)). An important aspect of our work -- that is relevant in its own right and of independent interest -- is that we identify connections between the string parameter of the locality number on the one hand, and the famous graph parameters of cutwidth and pathwidth, on the other hand. These two parameters have been jointly investigated in the literature and are arguably among the most central graph parameters that are based on "linearisations" of graphs. In this way, we also identify a direct approximation preserving reduction from cutwidth to pathwidth, which shows that any polynomial f(opt,|V|)-approximation algorithm for pathwidth yields a polynomial 2f(2 opt,h)-approximation algorithm for cutwidth on multigraphs (where h is the number of edges). In particular, this translates known approximation ratios for pathwidth into new approximation ratios for cutwidth, namely O(sqrt(log(opt)) log(h)) and O(sqrt(log(opt)) opt) for (multi) graphs with h edges.

翻译：本文研究局部性数这一近期提出的字符串结构参数（在带变量的模式匹配中有应用），及其与两个重要图参数——切割宽度和路径宽度之间的关联。这些关联使我们能够证明：计算局部性数是NP-难的，但若以局部性数或字母表大小为参数化变量，则属于固定参数可解问题——这恰好是文献中提出的开放性问题。此外，局部性数可实现比率为O(sqrt(log(opt)) log(n))的近似。本研究的一个重要方面（其本身具有独立研究价值）在于：我们揭示了字符串参数局部性数与著名图参数切割宽度和路径宽度之间的内在联系。这两个图参数在文献中常被共同研究，且可被视为基于图"线性化"的最核心图参数。通过这一关联，我们还建立了一个从切割宽度到路径宽度的直接近似保持归约，表明任何路径宽度的多项式f(opt,|V|)近似算法均可导出多图切割宽度的多项式2f(2 opt,h)近似算法（其中h为边数）。特别地，这可将已知的路径宽度近似比转化为切割宽度的新近似比：对含h条边的（多重）图，分别取得O(sqrt(log(opt)) log(h))和O(sqrt(log(opt)) opt)的近似比。