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Computational complexity of necessary envy-freeness – new study in Mathematical Social Sciences

by Haris Aziz, Ildikó Schlotte and Toby Walsh

 

Computational complexity of necessary envy-freeness

Haris Aziz, Ildikó Schlotter, Toby Walsh

Mathematical Social Sciences – Volume 127, January 2024, Pages 86-98

 

Highlights

 

  • Envy-free allocation of indivisible items to a few agents is computationally hard.
  • Finding a necessarily envy-free allocation is NP-hard already for 3 agents.
  • For any n ≥ 3 , finding a necessarily envy-free allocation for agents is NP-hard.

 

 

Abstract

We consider the fundamental problem of fairly allocating indivisible items when agents have strict ordinal preferences over individual items. We focus on the well-studied fairness criterion of necessary envy-freeness. For a constant number of agents, the computational complexity of the deciding whether there exists an allocation that satisfies necessary envy-freeness has been open for several years. We settle this question by showing that the problem is NP-complete even for three agents. Considering that the problem is polynomial-time solvable for the case of two agents, we provide a clear understanding of the complexity of the problem with respect to the number of agents.

Keywords: Fair division, Envy-freeness

 

 

 

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