Separator-Based Graph Embedding into Multidimensional Grids with Small Edge-Congestion

TL;DR

This paper presents a method for embedding graphs into multidimensional grids with minimal edge-congestion, leveraging graph separators to optimize the process based on the graph's properties.

Contribution

It introduces a new embedding technique that uses edge-separators with bounded expansion, achieving near-optimal edge-congestion for graphs with small separators.

Findings

Embedding with constant edge-congestion for graphs with small separators in higher dimensions.

Achieves $O( ext{degree})$ edge-congestion for graphs with fixed treewidth or excluded minors.

Provides an algorithm to construct edge-separators with bounded expansion.

Abstract

We study the problem of embedding a guest graph with minimum edge-congestion into a multidimensional grid with the same size as that of the guest graph. Based on a well-known notion of graph separators, we show that an embedding with a smaller edge-congestion can be obtained if the guest graph has a smaller separator, and if the host grid has a higher but constant dimension. Specifically, we prove that any graph with $N$ nodes, maximum node degree $\Delta$, and with a node-separator of size $O(n^\alpha)$ ($0\leq\alpha<1$) can be embedded into a grid of a fixed dimension $d\geq 2$ with at least $N$ nodes, with an edge-congestion of $O(\Delta)$ if $d>1/(1-\alpha)$, $O(\Delta\log N)$ if $d=1/(1-\alpha)$, and $O(\Delta N^{\alpha-1+\frac{1}{d}})$ if $d< 1/(1-\alpha)$. This edge-congestion achieves constant ratio approximation if $d>1/(1-\alpha)$, and matches an existential lower bound within a constant factor if $d\leq 1/(1-\alpha)$. Our result implies that if the guest graph has an excluded minor of a fixed size, such as a planar graph, then we can obtain an edge-congestion of $O(\Delta\log N)$ for $d=2$ and $O(\Delta)$ for any fixed $d\geq 3$. Moreover, if the guest graph has a fixed treewidth, such as a tree, an outerplanar graph, and a series-parallel graph, then we can obtain an edge-congestion of $O(\Delta)$ for any fixed $d\geq 2$. To design our embedding algorithm, we introduce edge-separators bounding expansion, such that in partitioning a graph into isolated nodes using edge-separators recursively, the number of outgoing edges from a subgraph to be partitioned in a recursive step is bounded. We present an algorithm to construct an edge-separator with expansion of $O(\Delta n^\alpha)$ from a node-separator of size $O(n^\alpha)$.