[Paper Review] Approximating the Cubicity of Trees

Jasine Babu, Manu Basavaraju+3 more|arXiv (Cornell University)|Feb 25, 2014

Advanced Graph Theory Research2 references3 citations

TL;DR

This paper presents the first constant-factor approximation algorithm for computing the cubicity of trees, both in randomized and deterministic polynomial time. It uses a recursive decomposition of trees and randomized weight-vector assignments to hypercubes, achieving a cube representation within a constant factor of the optimal dimension, with the randomized algorithm running in expected polynomial time and the deterministic version approximating cubicity without constructing the representation.

ABSTRACT

Cubicity of a graph $G$ is the smallest dimension $d$, for which $G$ is a unit disc graph in ${\mathbb{R}}^d$, under the $l^\infty$ metric, i.e. $G$ can be represented as an intersection graph of $d$-dimensional (axis-parallel) unit hypercubes. We call such an intersection representation a $d$-dimensional cube representation of $G$. Computing cubicity is known to be inapproximable in polynomial time, within an $O(n^{1-ε})$ factor for any $ε>0$, unless NP=ZPP. In this paper, we present a randomized algorithm that runs in polynomial time and computes cube representations of trees, of dimension within a constant factor of the optimum. It is also shown that the cubicity of trees can be approximated within a constant factor in deterministic polynomial time, if the cube representation is not required to be computed. As far as we know, this is the first constant factor approximation algorithm for computing the cubicity of trees. It is not yet clear whether computing the cubicity of trees is NP-hard or not.

Motivation & Objective

• To develop a polynomial-time approximation algorithm for the cubicity of trees, a parameter that measures the minimum dimension of unit hypercube intersection representations.
• To provide a randomized algorithm that computes a cube representation of a tree within a constant factor of the optimal dimension.
• To show that cubicity of trees can be approximated within a constant factor in deterministic polynomial time, even without explicitly computing the representation.
• To extend techniques from intrinsic dimensionality approximation to cubicity, despite the two parameters being fundamentally different.

Proposed method

• The algorithm recursively decomposes a tree into smaller subtrees, using a hierarchical decomposition based on depth and distance thresholds.
• For each subtree, it assigns weight vectors to edges using a randomized procedure that ensures non-adjacent vertices in the subtree are separated in at least one dimension.
• The randomized weight assignment uses a probability bound derived from the $ l^\infty $-norm of differences in projected coordinates, ensuring that non-adjacent pairs are not connected in the intersection graph.
• The algorithm applies a union bound over distance ranges to bound the failure probability of separating all non-adjacent vertex pairs.
• The number of dimensions used is set to $ 4(\lceil 22.77 \times \rho(T) \rceil + 2) $, where $ \rho(T) $ is the tree's radius, ensuring a constant-factor approximation.
• A deterministic version of the algorithm approximates the cubicity without constructing the representation, relying on the same structural decomposition and computable bounds.

Experimental results

Research questions

• RQ1 Can the cubicity of trees be approximated within a constant factor in polynomial time?
• RQ2 Is there a randomized algorithm that computes a cube representation of a tree within a constant factor of the optimal dimension?
• RQ3 Can the cubicity of trees be approximated deterministically in polynomial time, even without constructing the representation?
• RQ4 Do techniques from intrinsic dimensionality approximation apply to cubicity, despite the parameters being fundamentally different?

Key findings

• The randomized algorithm computes a cube representation of a tree in expected polynomial time, with dimension at most $ 4(\lceil 22.77 \times \rho(T) \rceil + 2) $, which is within a constant factor of the optimal cubicity.
• The failure probability of the randomized weight assignment is bounded by 0.33, ensuring that non-adjacent vertex pairs are separated in at least one dimension with probability at least 0.67.
• The expected number of repetitions of the weight assignment step is at most $ 1/0.67 $, meaning the algorithm performs a constant number of iterations in expectation.
• The deterministic version of the algorithm approximates the cubicity of trees within a constant factor in polynomial time, without constructing the actual cube representation.
• The algorithm's core technique, borrowed from intrinsic dimensionality, is adapted to cubicity with problem-specific modifications, despite the two parameters being incomparable in general.
• The paper establishes that computing the cubicity of trees is not known to be NP-hard, and the proposed algorithm provides the first constant-factor approximation for this problem.