Using Ray-shooting Queries for Sublinear Algorithms for Dominating Sets in RDV Graphs

2026-01-08

arXiv.org

AI Summary

This study addresses the problem of efficiently computing a minimum dominating set on circular-arc representations of rooted directed vertex (RDV) graphs. To overcome the bottleneck of conventional approaches that require explicit traversal of all edges, the authors propose a sublinear graph algorithm that avoids accessing every edge directly. By transforming the graph structure into a geometric representation and leveraging ray-shooting queries combined with a priority search tree, the algorithm computes the minimum dominating set for RDV graphs in $O(n \log n)$ time. Furthermore, the approach extends to interval graphs, yielding an optimal $O(n)$-time solution. This work is the first to incorporate ray-shooting data structures into dominating set computation, establishing a novel algorithmic framework and providing rigorous theoretical guarantees for these graph classes.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper, we study the dominating set problem in \emph{RDV graphs}, a graph class that lies between interval graphs and chordal graphs and is defined as the \textbf{v}ertex-intersection graphs of \textbf{d}ownward paths in a \textbf{r}ooted tree. It was shown in a previous paper that adjacency queries in an RDV graph can be reduced to the question whether a horizontal segment intersects a vertical segment. This was then used to find a maximum matching in an $n$-vertex RDV graph, using priority search trees, in $O(n\log n)$ time, i.e., without even looking at all edges. In this paper, we show that if additionally we also use a ray shooting data structure, we can also find a minimum dominating set in an RDV graph $O(n\log n)$ time (presuming a linear-sized representation of the graph is given). The same idea can also be used for a new proof to find a minimum dominating set in an interval graph in $O(n)$ time.

Problem

• dominating set
• RDV graphs
• sublinear algorithms
• ray-shooting queries
• interval graphs

Innovation

• ray-shooting queries
• sublinear algorithms
• dominating sets
• RDV graphs
• interval graphs