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Discrepancy Beyond Additive Functions with Applications to Fair Division (Extended Abstract)

Authors Alexandros Hollender , Pasin Manurangsi , Raghu Meka , Warut Suksompong

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Applied computing → Economics
Keywords
  • Discrepancy Theory
  • Fair Division

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Abstract

We consider a setting where we have a ground set ℳ together with real-valued set functions f₁, … , f_n, and the goal is to partition ℳ into two sets S₁,S₂ such that |f_i(S₁) - f_i(S₂)| is small for every i. Many results in discrepancy theory can be stated in this form with the functions f_i being additive. In this work, we initiate the study of the unstructured case where f_i is not assumed to be additive. We show that even without the additivity assumption, the upper bound remains at most O(√{n log n}). 
Our result has implications on the fair allocation of indivisible goods. In particular, we show that a consensus halving up to O(√{n log n}) goods always exists for n agents with monotone utilities. Previously, only an O(n) bound was known for this setting.

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Alexandros Hollender, Pasin Manurangsi, Raghu Meka, and Warut Suksompong. Discrepancy Beyond Additive Functions with Applications to Fair Division (Extended Abstract). In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, p. 77:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.77

Author Details

Alexandros Hollender
  • University of Oxford, UK
Pasin Manurangsi
  • Google Research, Bangkok, Thailand
Raghu Meka
  • University of California, Los Angeles, CA, USA
Warut Suksompong
  • National University of Singapore, Singapore

Funding

  • Meka, Raghu: Supported by the NSF Award CCF-2217033 (EnCORE: Institute for Emerging CORE Methods in Data Science).
  • Suksompong, Warut: Supported by the Singapore Ministry of Education under grant number MOE-T2EP20221-0001 and by an NUS Start-up Grant.
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