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Parameterized Complexity of Incomplete Connected Fair Division

Authors Harmender Gahlawat , Meirav Zehavi

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Author Details

Harmender Gahlawat
  • Ben-Gurion University of the Negev, Beersheba, Israel
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beersheba, Israel

Funding

This research was supported by the European Research Council (ERC) grant titled PARAPATH.

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Harmender Gahlawat and Meirav Zehavi. Parameterized Complexity of Incomplete Connected Fair Division. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.FSTTCS.2023.14

Abstract

Fair division of resources among competing agents is a fundamental problem in computational social choice and economic game theory. It has been intensively studied on various kinds of items (divisible and indivisible) and under various notions of fairness. We focus on Connected Fair Division (CFD), the variant of fair division on graphs, where the resources are modeled as an item graph. Here, each agent has to be assigned a connected subgraph of the item graph, and each item has to be assigned to some agent. 
We introduce a generalization of CFD, termed Incomplete CFD (ICFD), where exactly p vertices of the item graph should be assigned to the agents. This might be useful, in particular when the allocations are intended to be "economical" as well as fair. We consider four well-known notions of fairness: PROP, EF, EF1, EFX. First, we prove that EF-ICFD, EF1-ICFD, and EFX-ICFD are W[1]-hard parameterized by p plus the number of agents, even for graphs having constant vertex cover number (vcn). In contrast, we present a randomized FPT algorithm for PROP-ICFD parameterized only by p. Additionally, we prove both positive and negative results concerning the kernelization complexity of ICFD under all four fairness notions, parameterized by p, vcn, and the total number of different valuations in the item graph (val).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • Fair Division
  • Kernelization
  • Connected Fair Allocation
  • Fixed parameter tractability

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