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

## TL;DR

This paper explores the connections between the string parameter locality number and graph parameters cutwidth and pathwidth, establishing complexity results, approximation algorithms, and reductions that bridge string and graph theory.

## Contribution

It introduces the first fixed-parameter tractability results for locality number, relates it to cutwidth and pathwidth, and provides approximation-preserving reductions between these parameters.

## Key findings

- Computing the locality number is NP-hard but fixed-parameter tractable.
- The locality number can be approximated within a ratio of O(sqrt(log(opt)) log(n)).
- A reduction from cutwidth to pathwidth yields new approximation ratios for cutwidth.

## Abstract

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.