Document

# An ETH-Tight Algorithm for Multi-Team Formation

## File

LIPIcs.FSTTCS.2021.28.pdf
• Filesize: 0.67 MB
• 9 pages

## Acknowledgements

We would like to thank an anonymous reviewer for pointing us to the statement of [Kouteckỳ et al., 2018], allowing us to drastically simplify a previous version of the paper.

## Cite As

Daniel Lokshtanov, Saket Saurabh, Subhash Suri, and Jie Xue. An ETH-Tight Algorithm for Multi-Team Formation. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 28:1-28:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.FSTTCS.2021.28

## Abstract

In the Multi-Team Formation problem, we are given a ground set C of n candidates, each of which is characterized by a d-dimensional attribute vector in ℝ^d, and two positive integers α and β satisfying α β ≤ n. The goal is to form α disjoint teams T₁,...,T_α ⊆ C, each of which consists of β candidates in C, such that the total score of the teams is maximized, where the score of a team T is the sum of the h_j maximum values of the j-th attributes of the candidates in T, for all j ∈ {1,...,d}. Our main result is an 2^{2^O(d)} n^O(1)-time algorithm for Multi-Team Formation. This bound is ETH-tight since a 2^{2^{d/c}} n^O(1)-time algorithm for any constant c > 12 can be shown to violate the Exponential Time Hypothesis (ETH). Our algorithm runs in polynomial time for all dimensions up to d = clog log n for a sufficiently small constant c > 0. Prior to our work, the existence of a polynomial time algorithm was an open problem even for d = 3.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• Team formation
• Parameterized algorithms
• Exponential Time Hypothesis

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Nikhil Bansal, Tim Oosterwijk, Tjark Vredeveld, and Ruben Van Der Zwaan. Approximating vector scheduling: almost matching upper and lower bounds. Algorithmica, 76(4):1077-1096, 2016.
2. Marek Cygan, Marcin Pilipczuk, and Michal Pilipczuk. Known algorithms for edge clique cover are probably optimal. SIAM J. Comput., 45(1):67-83, 2016.
3. Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Kouteckỳ, Asaf Levin, and Shmuel Onn. An algorithmic theory of integer programming. arXiv preprint, 2019. URL: http://arxiv.org/abs/1904.01361.
4. Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998.
5. Erin L. Fitzpatrick and Ronald G. Askin. Forming effective worker teams with multi-functional skill requirements. Computers & Industrial Engineering, 48(3):593-608, 2005.
6. Raymond Hemmecke, Shmuel Onn, and Lyubov Romanchuk. N-fold integer programming in cubic time. Mathematical Programming, 137(1-2):325-341, 2013.
7. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972.
8. Jon Kleinberg and Maithra Raghu. Team performance with test scores. In Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC '15, pages 511-528, 2015.
9. Dušan Knop and Martin Kouteckỳ. Scheduling meets n-fold integer programming. Journal of Scheduling, 21(5):493-503, 2018.
10. Martin Kouteckỳ, Asaf Levin, and Shmuel Onn. A parameterized strongly polynomial algorithm for block structured integer programs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018.
11. Theodoros Lappas, Kun Liu, and Evimaria Terzi. Finding a team of experts in social networks. In Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '09, pages 467-476, 2009.
12. Patrick R. Laughlin and Andrea B. Hollingshead. A theory of collective induction. Organizational Behavior and Human Decision Processes, 61(1):94-107, 1995.
13. Jon Lee, Maxim Sviridenko, and Jan Vondrák. Submodular maximization over multiple matroids via generalized exchange properties. Mathematics of Operations Research, 35(4):795-806, 2010.
14. Tomasz P. Michalak, Talal Rahwan, Edith Elkind, Michael J. Wooldridge, and Nicholas R. Jennings. A hybrid exact algorithm for complete set partitioning. Artif. Intell., 230:14-50, 2016.
15. George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher. An analysis of approximations for maximizing submodular set functions - I. Math. Program., 14(1):265-294, 1978.
16. Scott Page. The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools, and Societies. Princeton University Press, 2007.
17. Marcin Pilipczuk and Manuel Sorge. A double exponential lower bound for the distinct vectors problem. CoRR, abs/2002.01293, 2020. URL: http://arxiv.org/abs/2002.01293.
18. Habibur Rahman, Senjuti Basu Roy, Saravanan Thirumuruganathan, Sihem Amer-Yahia, and Gautam Das. Optimized group formation for solving collaborative tasks. The VLDB Journal, 28(1):1-23, February 2019.
19. Talal Rahwan, Tomasz P. Michalak, Michael J. Wooldridge, and Nicholas R. Jennings. Coalition structure generation: A survey. Artif. Intell., 229:139-174, 2015.
20. Thomas Schibler, Ambuj Singh, and Subhash Suri. On multi-dimensional team formation. In Proc. of the 31st Canadian Conference on Computational Geometry, pages 146-152, 2019.
21. Travis C. Service and Julie A. Adams. Coalition formation for task allocation: theory and algorithms. Autonomous Agents and Multi-Agent Systems, 22(2):225-248, March 2011.
22. Marjorie E. Shaw. A comparison of individuals and small groups in the rational solution of complex problems. The American Journal of Psychology, 44(3):491-504, 1932.
23. Onn Shehory and Sarit Kraus. Methods for task allocation via agent coalition formation. Artif. Intell., 101(1-2):165-200, 1998.
24. I. D. Steiner. Group process and productivity. New York: Academic Press, 1972.
25. Xinyu Wang, Zhou Zhao, and Wilfred Ng. A comparative study of team formation in social networks. In Database Systems for Advanced Applications - 20th International Conference, DASFAA 2015, pages 389-404. Springer, 2015.