The Complexity of Vector Partition

TL;DR

This paper investigates the computational complexity of the vector partition problem, which involves dividing agents with attribute vectors into parts to minimize a cost function, with implications for clustering and logistics.

Contribution

It analyzes the problem's complexity and parameterized complexity under various parameters, highlighting open problems and theoretical challenges.

Findings

Complexity results depend on parameters p, d, a, t.

Identifies cases where the problem is computationally hard.

Raises open questions for future research.

Abstract

We consider the {\em vector partition problem}, where $n$ agents, each with a $d$-dimensional attribute vector, are to be partitioned into $p$ parts so as to minimize cost which is a given function on the sums of attribute vectors in each part. The problem has applications in a variety of areas including clustering, logistics and health care. We consider the complexity and parameterized complexity of the problem under various assumptions on the natural parameters $p,d,a,t$ of the problem where $a$ is the maximum absolute value of any attribute and $t$ is the number of agent types, and raise some of the many remaining open problems.