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When Far Is Better: The Chamberlin-Courant Approach to Obnoxious Committee Selection

Authors Sushmita Gupta , Tanmay Inamdar , Pallavi Jain , Daniel Lokshtanov , Fahad Panolan , Saket Saurabh

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Author Details

Sushmita Gupta
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Tanmay Inamdar
  • Indian Institute of Technology Jodhpur, Jodhpur, India
Pallavi Jain
  • Indian Institute of Technology Jodhpur, Jodhpur, India
Daniel Lokshtanov
  • University of California Santa Barbara, CA, USA
Fahad Panolan
  • School of Computer Science, University of Leeds, UK
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway

Funding

Sushmita Gupta and Pallavi Jain acknowledge support from SERB-SUPRA (grant number SPR/2021/000860).
  • Gupta, Sushmita: The author acknowledges the support from SERB-MATRICS (grant number MTR/2021/000869).
  • Inamdar, Tanmay: The author acknowledges support from IITJ Research Initiation Grant (grant number I/RIG/TNI/20240072).
  • Jain, Pallavi: The author acknowledges support from IITJ Seed Grant (grant number I/SEED/PJ/ 20210119).
  • Lokshtanov, Daniel: The author acknowledges support from NSF award CCF-2008838.
  • Saurabh, Saket: The author acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416); and he also acknowledges the support of Swarnajayanti Fellowship (grant number DST/SJF/MSA-01/2017-18).

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Sushmita Gupta, Tanmay Inamdar, Pallavi Jain, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. When Far Is Better: The Chamberlin-Courant Approach to Obnoxious Committee Selection. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 24:1-24:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.24

Abstract

Classical work on metric space based committee selection problem interprets distance as "near is better". In this work, motivated by real-life situations, we interpret distance as "far is better". Formally stated, we initiate the study of "obnoxious" committee scoring rules when the voters' preferences are expressed via a metric space. To accomplish this, we propose a model where large distances imply high satisfaction (in contrast to the classical setting where shorter distances imply high satisfaction) and study the egalitarian avatar of the well-known Chamberlin-Courant voting rule and some of its generalizations. For a given integer value λ between 1 and k, the committee size, a voter derives satisfaction from only the λth favorite committee member; the goal is to maximize the satisfaction of the least satisfied voter. For the special case of λ = 1, this yields the egalitarian Chamberlin-Courant rule.  In this paper, we consider general metric space and the special case of a d-dimensional Euclidean space. 
We show that when λ is 1 and k, the problem is polynomial-time solvable in ℝ² and general metric space, respectively. However, for λ = k-1, it is NP-hard even in ℝ². Thus, we have "double-dichotomy" in ℝ² with respect to the value of λ, where the extreme cases are solvable in polynomial time but an intermediate case is NP-hard. Furthermore, this phenomenon appears to be "tight" for ℝ² because the problem is NP-hard for general metric space, even for λ = 1. Consequently, we are motivated to explore the problem in the realm of (parameterized) approximation algorithms and obtain positive results. Interestingly, we note that this generalization of Chamberlin-Courant rules encodes practical constraints that are relevant to solutions for certain facility locations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
  • Theory of computation → Fixed parameter tractability
Keywords
  • Metric Space
  • Parameterized Complexity
  • Approximation
  • Obnoxious Facility Location

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