[Paper Review] Representing graphs as the intersection of axis-parallel cubes

Sunil L Chandran, Mathew C. Francis+1 more|ArXiv.org|Jul 19, 2006

Advanced Graph Theory Research30 references3 citations

TL;DR

This paper presents efficient algorithms to represent graphs as intersections of axis-parallel cubes, achieving a cubicity of $O(\Delta \ln n)$ and $O(\Delta \ln b)$, where $\Delta$ is the maximum degree and $b$ is the bandwidth. The approach uses bandwidth ordering and derandomized interval representations, with bounds tight up to $O(\log \log n)$ factors.

ABSTRACT

A unit cube in $k$ dimensional space (or \emph{$k$-cube} in short) is defined as the Cartesian product $R_1 imes R_2 imes... imes R_k$ where $R_i$(for $1\leq i\leq k$) is a closed interval of the form $[a_i,a_i+1]$ on the real line. A $k$-cube representation of a graph $G$ is a mapping of the vertices of $G$ to $k$-cubes such that two vertices in $G$ are adjacent if and only if their corresponding $k$-cubes have a non-empty intersection. The \emph{cubicity} of $G$, denoted as $\cubi(G)$, is the minimum $k$ such that $G$ has a $k$-cube representation. Roberts \cite{Roberts} showed that for any graph $G$ on $n$ vertices, $\cubi(G)\leq 2n/3$. Many NP-complete graph problems have polynomial time deterministic algorithms or have good approximation ratios in graphs of low cubicity. In most of these algorithms, computing a low dimensional cube representation of the given graph is usually the first step. We present an efficient algorithm to compute the $k$-cube representation of $G$ with maximum degree $Δ$ in $O(Δ\ln b)$ dimensions where $b$ is the bandwidth of $G$. Bandwidth of $G$ is at most $n$ and can be much lower. The algorithm takes as input a bandwidth ordering of the vertices in $G$. Though computing the bandwidth ordering of vertices for a graph is NP-hard, there are heuristics that perform very well in practice. Even theoretically, there is an $O(\log^4 n)$ approximation algorithm for computing the bandwidth ordering of a graph using which our algorithm can produce a $k$-cube representation of any given graph in $k=O(Δ(\ln b + \ln\ln n))$ dimensions. Both the bounds on cubicity are shown to be tight upto a factor of $O(\log\log n)$.

Motivation & Objective

• To develop efficient algorithms for representing arbitrary graphs as intersections of axis-parallel cubes in low-dimensional space.
• To reduce the cubicity of a graph—defined as the minimum dimension required for such a representation—based on structural graph parameters like maximum degree and bandwidth.
• To provide a geometric embedding of graphs into $\mathbb{R}^k$ such that adjacency corresponds exactly to $L_\infty$-distance at most 1.
• To improve upon the known upper bound of $2n/3$ for cubicity by leveraging graph parameters like bandwidth and degree.
• To design a polynomial-time algorithm that produces a low-dimensional cube representation using bandwidth ordering, even when bandwidth computation is NP-hard.

Proposed method

• Uses a derandomized algorithm (DET) to generate unit interval representations of indifference graphs derived from graph blocks.
• Applies a block decomposition strategy where vertices are partitioned into blocks based on bandwidth ordering, with each block processed independently.
• Constructs $k$-cube representations by combining interval representations across multiple layers using modulo indexing ($i = m \mod 3$) to ensure edge coverage and non-edge exclusion.
• Employs a supergraph construction $I_{i,j}$ for $0 \leq i \leq 2$, $1 \leq j \leq t$, where $t = \lceil \ln(2b) \rceil$, to represent the full graph as an intersection of indifference graphs.
• Derandomizes the construction using a deterministic procedure to assign intervals to vertices such that non-adjacent pairs are separated by more than $n$ units in at least one dimension.
• Combines the interval representations across dimensions to form a $k$-cube representation with $k = O(\Delta \ln b)$, using the bandwidth $b$ as a key parameter to reduce dimensionality.

Experimental results

Research questions

• RQ1 Can the cubicity of a graph be bounded more tightly using its maximum degree $\Delta$ and bandwidth $b$?
• RQ2 Is it possible to construct a low-dimensional $k$-cube representation of a graph in polynomial time using bandwidth ordering?
• RQ3 How does the cubicity of a graph relate to its bandwidth and maximum degree, and can tighter bounds be proven?
• RQ4 To what extent can the $O(\Delta \ln n)$ bound on cubicity be improved by incorporating bandwidth information?
• RQ5 Are the proposed bounds on cubicity tight up to logarithmic factors, particularly $O(\log \log n)$?

Key findings

• The paper presents an algorithm that computes a $k$-cube representation of any graph with maximum degree $\Delta$ in $O(\Delta \ln n)$ dimensions.
• A refined algorithm achieves a $k$-cube representation in $O(\Delta \ln b)$ dimensions, where $b$ is the bandwidth of the graph, improving upon the $O(\Delta \ln n)$ bound when $b \ll n$.
• Using an $O(\log^4 n)$-approximation algorithm for bandwidth, the method produces a $k$-cube representation in $k = O(\Delta(\ln b + \ln \ln n))$ dimensions.
• The bounds on cubicity are shown to be tight up to a factor of $O(\log \log n)$, as demonstrated by a tight example using a complete binary tree.
• For a complete binary tree of height $\log n$, the cubicity is $\Omega(\log n / \log \log n)$, and the upper bound is $O(\Delta \ln n) = O(\log n)$, confirming the tightness up to $O(\log \log n)$.
• The algorithm runs in polynomial time, relying on a derandomized construction of interval representations that ensures non-adjacent vertex pairs are separated in at least one dimension.