Graph Search Trees and the Intermezzo Problem

TL;DR

This paper proves the NP-completeness of the last-in-tree recognition problem for Generic Search and explores the complexity of the intermezzo problem, providing algorithms and hardness results related to partial orders.

Contribution

It establishes NP-completeness for the last-in-tree recognition problem and analyzes the intermezzo problem's complexity, offering algorithms and optimality results.

Findings

NP-completeness of last-in-tree recognition for Generic Search

NP-completeness of the intermezzo problem on tree-shaped partial orders

An XP-algorithm for the intermezzo problem parameterized by partial order width

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 $\pi$ and a set of triples, the $\mathsf{NP}$-complete intermezzo problem asks for a linear extension of $\pi$ 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.