Complexity of Last-in-Tree Recognition for Generic Graph Search and its Application to the Intermezzo Problem

Core Concepts

The last-in-tree recognition problem for Generic Search is NP-complete, and this result is used to show that the Intermezzo Problem remains NP-complete even when the partial order forms a tree of bounded height.

Abstract

The paper investigates the complexity of the last-in-tree recognition problem for Generic Search (GS) and its connection to the Intermezzo Problem.
Key insights:

The last-in-tree recognition problem for GS is NP-complete, even when restricted to rooted spanning trees of height 5. This is surprising as other graph search problems for GS are solvable in polynomial time.
The NP-completeness of the last-in-tree recognition problem for GS is used to show that the Intermezzo Problem remains NP-complete even when the partial order forms a cs-tree (a tree-like partial order) of bounded height.
In contrast, an XP algorithm is provided for the Intermezzo Problem when parameterized by the width of the partial order, and it is shown that this algorithm is asymptotically optimal under the Exponential Time Hypothesis.

The paper first introduces the necessary concepts and definitions, including graph searches, partial orders, and the last-in-tree recognition problem. It then proves the NP-completeness of the last-in-tree recognition problem for GS and uses this result to establish the NP-completeness of the Intermezzo Problem for cs-trees of bounded height. Finally, it presents the XP algorithm for the Intermezzo Problem parameterized by the width of the partial order.

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Key Insights Distilled From

Graph Search Trees and the Intermezzo Problem

by Jess... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18645.pdf

Deeper Inquiries

What are the potential applications of the NP-completeness result for the last-in-tree recognition problem of Generic Search

The NP-completeness result for the last-in-tree recognition problem of Generic Search has several potential applications in various fields. One application could be in network optimization, where understanding the complexity of recognizing certain tree structures in graph searches can lead to more efficient algorithms for network routing and design. Additionally, in computational biology, this result could be utilized in phylogenetic tree reconstruction, where identifying the correct tree structure is crucial for understanding evolutionary relationships. Furthermore, in social network analysis, recognizing specific tree patterns in graph searches can provide insights into information flow and influence propagation within networks.

Can the XP algorithm for the Intermezzo Problem parameterized by the width of the partial order be further improved, or is it asymptotically optimal

The XP algorithm for the Intermezzo Problem parameterized by the width of the partial order may not be further improvable, as it is already asymptotically optimal under the assumption of the Exponential Time Hypothesis. This hypothesis suggests that certain problems require exponential time to solve, and if the XP algorithm achieves this optimal running time, there may not be room for significant improvement. However, further research could focus on practical implementations and optimizations of the algorithm to enhance its efficiency in real-world applications.

Are there any other interesting connections between graph search problems and constrained ordering problems that could be explored

There are several interesting connections between graph search problems and constrained ordering problems that could be further explored. One potential avenue is investigating the relationship between different types of graph search trees and specific constraints on linear orderings, such as the betweenness problem and cyclic ordering problem. Understanding how these constraints interact with graph structures can provide insights into the complexity of recognizing certain tree patterns in graph searches. Additionally, exploring the connections between order theory and graph algorithms can lead to the development of novel algorithms for solving combinatorial optimization problems with constrained orderings.