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 (with respect to exact, parameterised and approximation algorithms), and are arguably among the most central graph parameters that are based on "linearisations" of graphs. Most importantly, we relate cutwidth with pathwidth via the locality number, which results in an approximation preserving reduction that improves the currently best known approximation algorithm for cutwidth.
[This is based on joint work with Katrin Casel, Joel D. Day, Pamela Fleischmann, Tomasz Kociumaka, and Florin Manea, published in Proc. ICALP'19.]