Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number

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 for-mulated as open problems in the literature. Moreover, the locality number can be approximated with ratio O (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.