The Parameterized Complexity of Clustering Incomplete Data

TL;DR

This paper investigates the parameterized complexity of clustering incomplete data, providing tight characterizations and showing fixed-parameter tractability when considering combined parameters, with intractability results when parameters are dropped.

Contribution

It offers a comprehensive complexity analysis of clustering incomplete data, identifying conditions for fixed-parameter tractability based on key parameters.

Findings

Problems are fixed-parameter tractable with combined parameters k, r, and coverage.

Dropping any parameter leads to intractability.

Complete data problems are fixed-parameter tractable by k+r.

Abstract

We study fundamental clustering problems for incomplete data. Specifically, given a set of incomplete d-dimensional vectors (representing rows of a matrix), the goal is to complete the missing vector entries in a way that admits a partitioning of the vectors into at most $k$ clusters with radius or diameter at most r. We give tight characterizations of the parameterized complexity of these problems with respect to the parameters k, r, and the minimum number of rows and columns needed to cover all the missing entries. We show that the considered problems are fixed-parameter tractable when parameterized by the three parameters combined, and that dropping any of the three parameters results in parameterized intractability. A byproduct of our results is that, for the complete data setting, all problems under consideration are fixed-parameter tractable parameterized by k+r.