Hyperbolic intersection graphs and (quasi)-polynomial time

Sándor Kisfaludi-Bak

AI summary · arXiv: 1812.03960v1 · cs.CG

Hyperbolic intersection graphs and (quasi)-polynomial time

Sándor Kisfaludi-Bak

Original paper Submitted to arXiv on 10 Dec. 2018 25 pages, 3 figures License: NONEXCLUSIVE-DISTRIB 1.0

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The essentials

The paper explores unit ball graphs and noisy uniform ball graphs in $d$-dimensional hyperbolic space $\mathbb{H}^d.
Novel separator theorem shows that unit ball graphs in $\mathbb{H}^d$ behave similarly to Euclidean counterparts but in a dimension one lower.
Significant advancements made in solving standard graph problems like Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle.
For fixed $d\geq 3$, problems can be solved in $2^{O(n^{1-1/(d-1)})}$ time in $\mathbb{H}^d compared to $2^{O(n^{1-1/d})}$ time in Euclidean space $\mathbb{R}^d.
Algorithms developed for $\mathbb{H}^d$ are optimal up to constant factors under ETH.
Notable achievement of quasi-polynomial algorithms for all studied problems in $\mathbb{H}^2, including polynomial time algorithms for Hamiltonian Cycle and 3-Coloring.
Efficiency of solving all analyzed problems efficiently in polynomial time if underlying noisy disks have constant ply in $\mathbb{H}^2.
Complementary quasi-polynomial algorithm presented for Independent Set with matching lower bound under ETH, marking the first instance of a proven tight quasi-polynomial lower bound.
Research highlights advantages of working with hyperbolic spaces for graph-related computational challenges and demonstrates how leveraging geometric properties can lead to significant algorithmic improvements.

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AI-generated keywords: Hyperbolic intersection graphs unit ball graphs noisy uniform ball graphs separator theorem quasi-polynomial time

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Original abstract

The paper "Hyperbolic intersection graphs and (quasi)-polynomial time" by Sándor Kisfaludi-Bak delves into the study of unit ball graphs and noisy uniform ball graphs in $d$-dimensional hyperbolic space, denoted as $\mathbb{H}^d$. The authors utilize a novel separator theorem to demonstrate that unit ball graphs in $\mathbb{H}^d$ exhibit similar properties to their Euclidean counterparts but in a dimension one lower. This leads to significant advancements in solving various standard graph problems such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle. Remarkably, for any fixed $d\geq 3$, these problems can be solved in $2^{O(n^{1-1/(d-1)})}$ time in $\mathbb{H}^d$, while requiring $2^{O(n^{1-1/d})}$ time in Euclidean space $\mathbb{R}^d$. The algorithms developed for $\mathbb{H}^d$ are shown to be optimal up to constant factors in the exponent under ETH. The impact of this dimensional reduction is most pronounced in $\mathbb{H}^2$, where a new technique is introduced to bound the treewidth of noisy uniform disk graphs. This advancement results in quasi-polynomial ($n^{O(\log n)}$) algorithms for all studied problems. Particularly noteworthy is the achievement of polynomial time algorithms for Hamiltonian Cycle and 3-Coloring. Moreover, if the underlying noisy disks in $\mathbb{H}^2$ have constant ply, all analyzed problems can be solved efficiently in polynomial time. This stands in stark contrast to the complexity exhibited by these problems under ETH in constant-ply Euclidean unit disk graphs, which require $2^{Ω(\sqrt{n})}$ time. The paper also presents a complementary quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs along with a matching lower bound of $n^{Ω(\log n)}$ under ETH. Notably, this marks the first instance of a natural problem with a quasi-polynomial lower bound that is proven to be tight. Overall, the research conducted sheds light on the potential advantages of working with hyperbolic spaces when tackling graph-related computational challenges and showcases how leveraging geometric properties can lead to significant algorithmic improvements.

Introduction:

The study of graphs and their properties has been a fundamental area of research in computer science for decades. Graphs are used to model various real-world problems, and finding efficient algorithms to solve them is crucial for many applications. In recent years, there has been a growing interest in exploring non-Euclidean spaces as potential alternatives to the traditional Euclidean space $\mathbb{R}^d$. One such space that has gained significant attention is the $d$-dimensional hyperbolic space, denoted as $\mathbb{H}^d$. The paper "Hyperbolic intersection graphs and (quasi)-polynomial time" by Sándor Kisfaludi-Bak delves into this topic by studying unit ball graphs and noisy uniform ball graphs in $\mathbb{H}^d$.

Background:

Before we dive into the details of the paper, let us first understand what unit ball graphs and noisy uniform ball graphs are. A unit ball graph is a graph where each vertex represents a unit-radius disk in some metric space, and two vertices are connected if their corresponding disks intersect. Similarly, a noisy uniform ball graph is a graph where each vertex represents a disk with some noise added to its radius, again connected if they intersect.

In Euclidean space $\mathbb{R}^d$, these types of graphs have been extensively studied due to their relevance in various applications such as wireless sensor networks, facility location problems, and social network analysis. However, working with these types of graphs can be computationally challenging due to their high dimensionality.

Main Contributions:

The authors' main contribution lies in developing novel techniques that exploit the geometric properties of hyperbolic spaces to significantly improve upon existing algorithms for solving standard graph problems on unit ball graphs and noisy uniform ball graphs.

One key result presented in the paper is a separator theorem for unit ball graphs in $\mathbb{H}^d$. This theorem states that for any fixed $d\geq 3$, the treewidth of a unit ball graph in $\mathbb{H}^d$ is at most $n^{1-1/(d-1)}$, where $n$ is the number of vertices in the graph. This result is crucial as it shows that unit ball graphs in $\mathbb{H}^d$ behave similarly to their Euclidean counterparts but in a dimension one lower. As a consequence, this leads to significant advancements in solving various standard graph problems such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle.

The authors also introduce a new technique for bounding the treewidth of noisy uniform disk graphs in $\mathbb{H}^2$. This advancement results in quasi-polynomial ($n^{O(\log n)}$) algorithms for all studied problems on these types of graphs. Notably, this includes polynomial time algorithms for Hamiltonian Cycle and 3-Coloring. Moreover, if the underlying noisy disks have constant ply (i.e., bounded by some constant), all analyzed problems can be solved efficiently in polynomial time. This stands in stark contrast to their complexity under ETH (Exponential Time Hypothesis) on constant-ply Euclidean unit disk graphs, which require $2^{Ω(\sqrt{n})}$ time.

Another noteworthy contribution of this paper is presenting a quasi-polynomial algorithm for Independent Set on noisy uniform disk graphs along with a matching lower bound of $n^{Ω(\log n)}$ under ETH. What makes this result particularly interesting is that it marks the first instance of a natural problem with a quasi-polynomial lower bound that has been proven to be tight.

Conclusion:

In conclusion, "Hyperbolic intersection graphs and (quasi)-polynomial time" presents groundbreaking research on utilizing hyperbolic spaces to solve various standard graph problems efficiently. The paper's results demonstrate the potential advantages of working with hyperbolic spaces and highlight how leveraging geometric properties can lead to significant algorithmic improvements. This research opens up new avenues for exploring non-Euclidean spaces in solving computational challenges and paves the way for future advancements in this area.

Created on 09 Jul. 2026

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