An ETH-Tight Algorithm for Multi-Team Formation

Authors Daniel Lokshtanov, Saket Saurabh, Subhash Suri, Jie Xue

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Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA
Saket Saurabh
  • The Institute of Mathematical Sciences (HBNI), Chennai, India
  • University of Bergen, Norway
Subhash Suri
  • University of California, Santa Barbara, CA, USA
Jie Xue
  • New York University Shanghai, China


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.

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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)


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
  • Team formation
  • Parameterized algorithms
  • Exponential Time Hypothesis


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