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Beyond Exact Fairness: Envy-Free Incomplete Connected Fair Division

Authors Ajaykrishnan E S , Daniel Lokshtanov

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Subject Classification

ACM Subject Classification
  • Theory of computation → W hierarchy
  • Theory of computation → Fixed parameter tractability
Keywords
  • Envy-Free Incomplete Connected Fair Division
  • Efficient Parameterized Approximation Scheme
  • W[1]-hardness

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Abstract

We study the problem of Envy-Free Incomplete Connected Fair Division, where exactly p vertices of an undirected graph must be allocated to agents such that each agent receives a connected share and does not envy another agent’s share. Focusing on agents with additive valuations, we show that the problem remains computationally hard when parameterized by p and the number of agents. This result holds even for star graphs and with the input numbers given in unary representation, thereby resolving an open problem posed by Gahlawat and Zehavi (FSTTCS 2023).
In stark contrast, we show that if one is willing to tolerate even the slightest amount of envy, then the problem becomes efficient with respect to the natural parameters. Specifically, we design an Efficient Parameterized Approximation Scheme parameterized by p and the number of agent types. Our algorithm works on general graphs and remains efficient even when the input numbers are provided in binary representation.

Cite As Get BibTex

Ajaykrishnan E S and Daniel Lokshtanov. Beyond Exact Fairness: Envy-Free Incomplete Connected Fair Division. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.29

Author Details

Ajaykrishnan E S
  • University of California, Santa Barbara, CA, USA
Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA

Funding

NSF Grant No. 2505099, Collaborative Research: AF Medium: Structure and Quasi-Polynomial Time Algorithms.

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