Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number
Katrin Casel, Joel D. Day, Pamela Fleischmann, Tomasz Kociumaka, Florin Manea, Markus L. Schmid
TL;DR
This paper investigates the locality number $\operatorname{loc}(\alpha)$ of strings and its deep connections to two central graph layout parameters, cutwidth $\operatorname{cw}(G)$ and pathwidth $\operatorname{pw}(G)$. It establishes approximation-preserving reductions between these domains, proving that computing $\operatorname{loc}$ is NP-hard but fixed-parameter tractable when parameterised by the alphabet size $|\Sigma|$ or by $\operatorname{loc}(\alpha)$, and giving a $O(\sqrt{\log(\mathrm{opt})}\log n)$-approximation for MinLoc. The paper also shows how approximations for pathwidth translate into cutwidth approximations via new reductions, yielding concrete bounds such as $O(\sqrt{\log(\mathrm{opt})}\log h)$ and $O(\sqrt{\log(\mathrm{opt})}\mathrm{opt})$ for MinCutwidth on multigraphs, and introduces a second-order cutwidth $\operatorname{cw}_2$ to bridge cutwidth and pathwidth directly. Overall, these results fuse string combinatorics with classical graph parameters, enabling transfer of algorithmic techniques and closing several open questions about the locality parameter. The findings have practical relevance for pattern matching with variables and for leveraging graph-layout heuristics in string-processing problems.
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.