Graph Search Trees and the Intermezzo Problem
Jesse Beisegel, Ekkehard Köhler, Fabienne Ratajczak, Robert Scheffler, Martin Strehler
TL;DR
This paper investigates the complexity of last-in-tree recognition for Generic Search ($\mathsf{GS}$) and the Intermezzo total-ordering problem. It proves that the $\mathcal{L}$-Tree Recognition Problem under $\mathsf{GS}$ is $NP$-complete on general graphs, and uses this to establish $NP$-completeness of Intermezzo even when the poset is restricted to a cs-tree or bounded height. It also presents a width-parameterized $\mathsf{XP}$-algorithm for Intermezzo and shows ETH-based lower bounds on its running time. Overall, the results delineate a boundary between tractable end-vertex recognition for $\mathsf{GS}$ and intractable tree-recognition, linking graph-search theory with order-theoretic constraints.
Abstract
The last in-tree recognition problem asks whether a given spanning tree can be derived by connecting each vertex with its rightmost left neighbor of some search ordering. In this study, we demonstrate that the last-in-tree recognition problem for Generic Search is $\mathsf{NP}$-complete. We utilize this finding to strengthen a complexity result from order theory. Given a partial order $π$ and a set of triples, the $\mathsf{NP}$-complete intermezzo problem asks for a linear extension of $π$ where each first element of a triple is not between the other two. We show that this problem remains $\mathsf{NP}$-complete even when the Hasse diagram of the partial order forms a tree of bounded height. In contrast, we give an $\mathsf{XP}$-algorithm for the problem when parameterized by the width of the partial order. Furthermore, we show that $\unicode{x2013}$ under the assumption of the Exponential Time Hypothesis $\unicode{x2013}$ the running time of this algorithm is asymptotically optimal.