Geometric representation of graphs in low dimension

TL;DR

This paper presents an efficient randomized algorithm for constructing low-dimensional box representations of graphs, providing tight bounds related to maximum and average degrees, with implications for graph visualization and analysis.

Contribution

It introduces a new randomized algorithm for graph boxicity representation with tight bounds and a derandomized version, advancing understanding of graph dimension constraints.

Findings

Algorithm constructs box representations in $1.5 ( ext{max degree} + 2) ext{ln} n$ dimensions.

Upper bounds on boxicity are tight up to a factor of $ ext{ln} n$.

For most graphs, boxicity is bounded by a function of average degree and $ ext{ln} n$.

Abstract

We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in $1.5 (\Delta + 2) \ln n$ dimensions, where $\Delta$ is the maximum degree of G. We also show that $\boxi(G) \le (\Delta + 2) \ln n$ for any graph G. Our bound is tight up to a factor of $\ln n$. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree $\Delta$, we show that for almost all graphs on n vertices, its boxicity is upper bound by $c\cdot(d_{av} + 1) \ln n$ where d_{av} is the average degree and c is a small constant. Also, we show that for any graph G, $\boxi(G) \le \sqrt{8 n d_{av} \ln n}$, which is tight up to a factor of $b \sqrt{\ln n}$ for a constant b.