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Binary k-Center with Missing Entries: Structure Leads to Tractability

Authors Tobias Friedrich , Kirill Simonov , Farehe Soheil

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ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Clustering
  • Missing Entries
  • k-Center
  • Parameterized Algorithms

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Abstract

k-Center clustering is a fundamental classification problem, where the task is to categorize the given collection of entities into k clusters and come up with a representative for each cluster, so that the maximum distance between an entity and its representative is minimized. In this work, we focus on the setting where the entities are represented by binary vectors with missing entries, which model incomplete categorical data. This version of the problem has wide applications, from predictive analytics to bioinformatics.
Our main finding is that the problem, which is notoriously hard from the classical complexity viewpoint, becomes tractable as soon as the known entries are sparse and exhibit a certain structure. Formally, we show fixed-parameter tractable algorithms for the parameters vertex cover, fracture number, and treewidth of the row-column graph, which encodes the positions of the known entries of the matrix. Additionally, we tie the complexity of the 1-cluster variant of the problem, which is famous under the name Closest String, to the complexity of solving integer linear programs with few constraints. This implies, in particular, that improving upon the running times of our algorithms would lead to more efficient algorithms for integer linear programming in general.

Cite As Get BibTex

Tobias Friedrich, Kirill Simonov, and Farehe Soheil. Binary k-Center with Missing Entries: Structure Leads to Tractability. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.8

Author Details

Tobias Friedrich
  • Hasso Plattner Institute (HPI), University of Potsdam, Germany
Kirill Simonov
  • Department of Informatics, University of Bergen, Norway
Farehe Soheil
  • Hasso Plattner Institute (HPI), University of Potsdam, Germany

Funding

  • Soheil, Farehe: Supported by the HPI Research School on Foundations of AI (FAI).

Acknowledgements

We thank Nikolai Karpov for pointing out the applications in bioinformatics.

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