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Obtaining a Proportional Allocation by Deleting Items.

Britta Dorn1, Ronald de Haan2, Ildikó Schlotter3

  • 1University of Tübingen, Tübingen, Germany.

Algorithmica
|November 1, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces Proportionality by Item Deletion (PID), a problem focused on fairly allocating indivisible goods. A polynomial-time algorithm is presented for three agents, but the problem becomes intractable for more agents, highlighting complexity in fair division.

Keywords:
Computational complexityControlFair divisionItem deletionParameterized complexityProportional allocation

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Area of Science:

  • Algorithmic Game Theory
  • Computational Social Choice
  • Fair Division

Background:

  • Fair allocation of indivisible goods is a fundamental challenge in resource distribution.
  • Existing methods often struggle with computational complexity as the number of agents or items increases.

Purpose of the Study:

  • To investigate the computational complexity of ensuring proportional allocation of indivisible goods by deleting a minimum number of items.
  • To develop efficient algorithms for specific cases and establish hardness results for general instances.

Main Methods:

  • Introduced the Proportionality by Item Deletion (PID) problem.
  • Developed a polynomial-time algorithm for PID with three agents.
  • Proved NP-hardness for PID with an unbounded number of agents, parameterized by the number of deletions (k).
  • Analyzed approximation possibilities and studied a variant with a pre-defined allocation.

Main Results:

  • A polynomial-time algorithm for PID with three agents was achieved.
  • PID is proven to be computationally intractable for an unbounded number of agents, with k-parameterized hardness.
  • Strong inapproximability results for PID were established.
  • A variant of PID is NP-hard for six agents but polynomial for two, and k-parameterized hard.

Conclusions:

  • The computational complexity of fair allocation problems varies significantly with the number of agents.
  • Efficient solutions exist for small numbers of agents, but general instances pose significant computational challenges.
  • The study provides a comprehensive analysis of PID, including algorithmic, complexity, and approximation aspects.