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Group Activity Selection with Few Agent Types

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Abstract

In this paper we establish the complexity map for the Group Activity Selection Problem (GASP), along with two of its prominent variants called sGASP and gGASP, focusing on the case when the number of types of agents is the parameter. In all these problems, one is given a set of agents (each with their own preferences) and a set of activities, and the aim is to assign agents to activities in a way which satisfies certain global as well as preference-based conditions. Our positive results, consisting of one fixed-parameter algorithm and one XP algorithm, rely on a combination of novel Subset Sum machinery (which may be of general interest) and identifying certain compression steps that allow us to focus on solutions with a simpler, well-defined structure (in particular, they are “acyclic”). These algorithms are complemented by matching lower bounds, which among others close a gap to a recently obtained tractability result of Gupta et al. (in: Algorithmic game theory—10th international symposium, SAGT 2017, vol 10504 of lecture notes in computer science, Springer, 2017). In this direction, the techniques used to establish W[1]-hardness of sGASP are of particular interest: as an intermediate step, we use Sidon sequences to show the W[1]-hardness of a highly restricted variant of multi-dimensional Subset Sum, which may find applications in other settings as well.

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Notes

  1. To avoid any confusion, we stress that in line with previous work our model allows for ties.

  2. A formal definition is provided at the beginning of Sect. 4.

  3. Preference lists of this form are sometimes called Ballester encodings [2].

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Acknowledgements

Rahul C. S. acknowledges support by project TOTAL funded by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 677651). Robert Ganian acknowledges support from the Austrian Science Fund (FWF, projects Y1329 and P31336). Sebastian Ordyniak acknowledges support from the Engineering and Physical Sciences Research Council (EPSRC, project EP/V00252X/1).

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Authors and Affiliations

  1. TU Wien, Vienna, Austria

    Robert Ganian

  2. University of Leeds, Leeds, UK

    Sebastian Ordyniak

  3. IIT Goa, Goa, India

    C. S. Rahul

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  1. Robert Ganian
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  2. Sebastian Ordyniak
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  3. C. S. Rahul
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Correspondence to C. S. Rahul.

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A shortened and preliminary version of this paper has been published in the proceedings of the 27th Annual European Symposium on Algorithms (ESA 2019).

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Ganian, R., Ordyniak, S. & Rahul, C.S. Group Activity Selection with Few Agent Types. Algorithmica 85, 1111–1155 (2023). https://doi.org/10.1007/s00453-022-01058-z

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