Partitioning Vectors into Quadruples: Worst-Case Analysis of a Matching-Based Algorithm

TL;DR

This paper studies a problem of partitioning 4k vectors into quads to minimize total cost, analyzing a matching-based algorithm's worst-case performance and its effectiveness on special cases.

Contribution

It provides a worst-case approximation ratio analysis of a simple matching-based algorithm for the vector partitioning problem, including tight bounds for special cases.

Findings

The algorithm is a (3/2)-approximation in the general case.

In a specific special case, the algorithm achieves a (5/4)-approximation.

The analysis bounds are tight in all but one case.

Abstract

Consider a problem where 4k given vectors need to be partitioned into k clusters of four vectors each. A cluster of four vectors is called a quad, and the cost of a quad is the sum of the component-wise maxima of the four vectors in the quad. The problem is to partition the given 4k vectors into k quads with minimum total cost. We analyze a straightforward matching-based algorithm, and prove that this algorithm is a (3/2)-approximation algorithm for this problem. We further analyze the performance of this algorithm on a hierarchy of special cases of the problem, and prove that, in one particular case, the algorithm is a (5/4)-approximation algorithm. Our analysis is tight in all cases except one.