Homogeneous team formation is the task of grouping individuals into teams, each of which consists of members who fulfill the same set of prespecified properties. In this theoretical work, we propose, motivate, and analyze a combinatorial model where, given a matrix over a finite alphabet whose rows correspond to individuals and columns correspond to attributes of individuals, the user specifies lower and upper bounds on team sizes as well as combinations of attributes that have to be homogeneous (that is, identical) for all members of the corresponding teams. Furthermore, the user can define a cost for assigning any individual to a certain team. We show that some special cases of our new model lead to NP-hard problems while others allow for (fixed-parameter) tractability results. For example, the problem is already NP-hard even if (i) there are no lower and upper bounds on the team sizes, (ii) all costs are zero, and (iii) the matrix has only two columns. In contrast, the problem becomes fixed-parameter tractable for the combined parameter “number of possible teams” and “number of different individuals”, the latter being upper-bounded by the number of rows.
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Informally speaking, a problem with input size x and parameter p is called fixed-parameter tractable if it can be solved in f(p)⋅x O(1) time, where f may be an arbitrarily computable function solely depending on p.
Although the input table as well as the given patterns formally are matrices, we use different terms to distinguish between them: The “input matrix” consisting of “rows” and the “pattern mask” consisting of “pattern vectors”.
In (2) the modified Row Assignment ∗ has a specific lower bound α j and a specific upper bound β j for each j∈T out instead of a uniform upper bound k.
As a consequence of the “binarization” in Lemma 1, the question whether Homogeneous Team Formation is fixed-parameter tractable with respect to the combined parameter (p,|Σ|) is equivalent to the question whether Homogeneous Team Formation is fixed-parameter tractable with respect to p alone.
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We are grateful to the anonymous referees of the MFCS’11 conference for helping to improve this work by spotting some flaws and providing the idea behind Corollary 4. Furthermore, we thank an anonymous referee for providing the idea of the proof of Theorem 2 which is significantly simpler than the one in the conference version of this paper. We are also grateful to two anonymous Algorithmica reviewers for their constructive feedback.
R. Bredereck and T. Köhler were supported by the DFG, research project PAWS, NI 369/10.
Major parts of this work were done while G. Philip was with The Institute of Mathematical Sciences, Chennai, India, and visiting TU Berlin.
Parts of this work were done while T. Köhler was student at Friedrich-Schiller-Universität Jena and visiting TU Berlin as student research assistent.
An extended abstract appeared under the title “Pattern-Guided Data Anonymization and Clustering” in Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS’11), volume 6907 of LNCS, pages 182–193, Springer 2011. That version concentrates on the anonymization aspects of the model. In our new version we slightly extend our model and show how it applies to (homogeneous) clustering of individuals, that is, to homogeneous team formation. Indeed, we now claim that the models and ideas better fit with these applications than with the previous data anonymization motivation. Apart from full proofs omitted in the extended abstract and also adapting our old ideas to the new extended model, the current article also contains a new and easier proof of NP-hardness, a new proof for showing that polynomial-time data reduction in term of so-called polynomial-size problem kernels is unlikely to exist with respect to certain parameterizations, and a new algorithm for the (still NP-hard) special case ignoring costs. Many of the new findings are part of the diploma thesis  of Thomas Köhler.
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Bredereck, R., Köhler, T., Nichterlein, A. et al. Using Patterns to Form Homogeneous Teams. Algorithmica 71, 517–538 (2015). https://doi.org/10.1007/s00453-013-9821-0
- Team selection
- Team formation
- Matrix modification problems
- Parameterized complexity
- Fixed-parameter tractability