Multidimensional Scaling: Approximation and Complexity

TL;DR

This paper proves that minimizing the Kamada-Kawai objective in metric multidimensional scaling is NP-hard, and introduces a provable approximation algorithm, including a PTAS for low-diameter graphs, with experimental insights.

Contribution

It establishes the NP-hardness of Kamada-Kawai MDS and provides the first approximation algorithm with theoretical guarantees, including a PTAS for specific graph classes.

Findings

Kamada-Kawai objective minimization is NP-hard.

A provable approximation algorithm for the objective is developed.

A PTAS exists for low-diameter graphs.

Abstract

Metric Multidimensional scaling (MDS) is a classical method for generating meaningful (non-linear) low-dimensional embeddings of high-dimensional data. MDS has a long history in the statistics, machine learning, and graph drawing communities. In particular, the Kamada-Kawai force-directed graph drawing method is equivalent to MDS and is one of the most popular ways in practice to embed graphs into low dimensions. Despite its ubiquity, our theoretical understanding of MDS remains limited as its objective function is highly non-convex. In this paper, we prove that minimizing the Kamada-Kawai objective is NP-hard and give a provable approximation algorithm for optimizing it, which in particular is a PTAS on low-diameter graphs. We supplement this result with experiments suggesting possible connections between our greedy approximation algorithm and gradient-based methods.