On $G^p$-unimodality of radius functions in graphs: structure and algorithms

2025-03-19

AI Summary

This paper investigates the relationship between local minima of the weighted radius function and global centrality in graphs, introducing and characterizing the discrete unimodal property “$G^2$-unimodality”: if a vertex minimizes the radius function within its 2-neighborhood, it must be a graph center. This property generalizes classical Helly graph theory and yields the first structural characterization of $G^2$-unimodal graphs, which strictly contain weakly bridged graphs, convex-ball graphs, and bipartite Helly graphs. Leveraging combinatorial graph theory and extensions of the Helly property, we design near-linear-time algorithms for center computation on several important graph subclasses. Furthermore, via a reduction from the hitting set conjecture, we refute the existence of a universal $widetilde{O}(sqrt{n}m)$-time local search algorithm for center finding on this graph class, thereby establishing a tight complexity bound for local-search-based center computation.

Technology Category

• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

For every weight assignment $pi$ to the vertices in a graph $G$, the radius function $r_pi$ maps every vertex of $G$ to its largest weighted distance to the other vertices. The center problem asks to find a center, i.e., a vertex of $G$ that minimizes $r_pi$. We here study some local properties of radius functions in graphs, and their algorithmic implications; our work is inspired by the nice property that in Euclidean spaces every local minimum of every radius function $r_pi$ is a center. We study a discrete analogue of this property for graphs, which we name $G^p$-unimodality: specifically, every vertex that minimizes the radius function in its ball of radius $p$ must be a central vertex. While it has long been known since Dragan (1989) that graphs with $G$-unimodal radius functions $r_pi$ are exactly the Helly graphs, the class of graphs with $G^2$-unimodal radius functions has not been studied insofar. We prove the latter class to be much larger than the Helly graphs, since it also comprises (weakly) bridged graphs, graphs with convex balls, and bipartite Helly graphs. Recently, using the $G$-unimodality of radius functions $r_pi$, a randomized $widetilde{mathcal{O}}(sqrt{n}m)$-time local search algorithm for the center problem on Helly graphs was proposed by Ducoffe (2023). Assuming the Hitting Set Conjecture (Abboud et al., 2016), we prove that a similar result for the class of graphs with $G^2$-unimodal radius functions is unlikely. However, we design local search algorithms (randomized or deterministic) for the center problem on many of its important subclasses.

Problem

• Study local properties of radius functions in graphs.
• Explore algorithmic implications of $G^p$-unimodality in graphs.
• Design local search algorithms for center problem on specific graph classes.

Innovation

• Introduces $G^p$-unimodality for graph radius functions
• Proves $G^2$-unimodal graphs include weakly bridged graphs
• Designs local search algorithms for center problem