Researchers Prove Hitting Geodesic Intervals Remains NP-complete Even on Graphs with 5-vertex Paths

The challenge of finding the smallest set of vertices that connects all specified pairs within a network, known as Hitting Geodesic Intervals, receives detailed examination in new research led by Tatsuya Gima and Yasuaki Kobayashi of Hokkaido University, alongside Yuto Okada, Yota Otachi, and Hayato Takaike from Nagoya University. This study advances understanding of the problem’s complexity, demonstrating its inherent difficulty even within seemingly simple graph structures, and establishing limitations for efficient algorithms based on common graph parameters. The team proves the problem remains computationally intractable on graphs with limited bandwidth or degree, suggesting that solving large instances will likely require substantial computational resources, but also presents algorithms that efficiently solve the problem when parameterized by specific structural properties like plus modular-width and plus vertex integrity. These findings not only refine the theoretical landscape of network optimization but also offer valuable insights for designing practical solutions in areas such as communication networks and logistics.

Parameterized Algorithms for Geodesic Interval Problems

This body of work explores the computational challenges of solving the Hitting Geodesic Intervals (HGI) problem, which asks whether a small set of vertices can intersect all shortest paths between pairs of designated points in a network. The research focuses on parameterized complexity, a field that seeks efficient algorithms for problems that are generally difficult by considering specific characteristics of the input. The overarching goal is to identify parameters that, when combined with the problem size, allow for tractable solutions. The studies investigate various graph algorithms and techniques for solving hitting set problems, where the aim is to find a small set of elements that intersect a collection of sets.

Several papers examine the parameterized complexity of hitting set problems in general, exploring different parameters to achieve fixed-parameter tractability, a desirable property for algorithms. Research also delves into the complexity of finding geodetic sets and how graph structure, such as pathwidth and feedback vertex set number, impacts computational difficulty. The core of the research centers on the HGI problem itself, aiming to find small sets of vertices that intersect all shortest paths between pairs of points. The studies explore various parameters to achieve efficient algorithms for this challenging problem.

The primary goal of this research is to develop fixed-parameter algorithms for the HGI problem, meaning algorithms that run in polynomial time when combined with a specific parameter. The structure of the graph plays a significant role, and the research investigates how different structures can be exploited to develop more efficient algorithms. The HGI problem is closely related to other graph theory problems, such as multicut, graph separation, and hitting set, allowing researchers to leverage existing techniques and results. Understanding the relationships between different graph parameters is crucial for identifying the most effective parameters for achieving efficient algorithms.

Hitting Geodesic Intervals Remains NP-Complete

Researchers have established a clearer understanding of the computational complexity of the Hitting Geodesic Intervals (HGI) problem, which seeks a small set of vertices that intersect all shortest paths between pairs of points in a graph. The study rigorously defines the boundaries between solvable and intractable cases, focusing on how graph structure impacts computational difficulty. The research demonstrates that HGI remains NP-complete even when restricted to remarkably simple graph structures, including graphs formed by adding a single vertex to five-vertex paths, paths and triangles, and graphs with limited bandwidth and maximum degree. These findings suggest that fixed-parameter algorithms are unlikely to exist for commonly studied graph structures without significant breakthroughs in complexity theory.

However, the research also delivers positive results, identifying conditions under which HGI can be solved efficiently. The team developed fixed-parameter algorithms based on two structural graph parameters: k plus modular-width and k plus vertex integrity. Notably, the algorithm utilizing k plus vertex integrity solves a more general case, encompassing parameterization by the minimum vertex multiway-cut size of the terminal vertices. Further analysis demonstrates the tightness of this approach, proving that the problem remains W-complete when parameterized by the minimum vertex multicut size, even with limited terminal pairs. These findings provide a clearer understanding of the inherent complexity of HGI, offering valuable insights for researchers tackling related problems in network monitoring, facility location, and communication network design.

Hitting Geodesic Intervals Remains NP-complete

This research investigates the computational complexity of the Hitting Geodesic Intervals (HGI) problem, which seeks a small set of vertices that intersect all shortest paths between specified pairs of points in a graph. The study extends previous work by establishing both negative and positive results regarding the problem’s solvability. Researchers demonstrate that HGI remains NP-complete even when restricted to relatively simple graph structures, including graphs formed by adding a single vertex to disjoint paths or triangles, and graphs with limited bandwidth and maximum degree. However, the research also identifies conditions under which HGI can be solved efficiently.

The team proves that HGI is fixed-parameter tractable when parameterized by certain graph measures, specifically k plus modular-width and k plus vertex integrity, offering practical algorithmic approaches for specific graph types. Notably, the algorithm for k plus vertex integrity also addresses a more general problem involving minimum vertex multiway cuts. The study confirms the tightness of this result by demonstrating that HGI parameterized by minimum vertex multicut size is, in fact, W-complete. Furthermore, the research establishes the polynomial-time solvability of HGI on trees and draws connections to known results on Vertex Cover and geometric problems like Piercing Rectangles.