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Envy-Free and Efficient Allocations for Graphical Valuations

Neeldhara Misra, Aditi Sethia

TL;DR

It is shown that the price of EFX with respect to utilitarian welfare is one for binary utilities, but can be arbitrarily large {0, 1, d} valuations, but can be arbitrarily large {0, 1, d} valuations.

Abstract

We consider the complexity of finding envy-free allocations for the class of graphical valuations. Graphical valuations were introduced by Christodoulou et. al.(2023) as a structured class of valuations that admit allocations that are envy-free up to any item (EFX). These are valuations where every item is valued by two agents, lending a (simple) graph structure to the utilities, where the agents are vertices and are adjacent if and only if they value a (unique) common item. Finding envy-free allocations for general valuations is known to be computationally intractable even for very special cases: in particular, even for binary valuations, and even for identical valuations with two agents. We show that, for binary graphical valuations, the existence of envy-free allocations can be determined in polynomial time. In contrast, we also show that allowing for even slightly more general utilities {0, 1, d} leads to intractability even for graphical valuations. This motivates other approaches to tractability, and to that end, we exhibit the fixed-parameter tractability of the problem parameterized by the vertex cover number of the graph when the number of distinct utilities is bounded. We also show that, all graphical instances that admit EF allocations also admit one that is non-wasteful. Since EFX allocations are possibly wasteful, we also address the question of determining the price of fairness of EFX allocations. We show that the price of EFX with respect to utilitarian welfare is one for binary utilities, but can be arbitrarily large {0, 1, d} valuations. We also show the hardness of deciding the existence of an EFX allocation which is also welfare-maximizing and of finding a welfare-maximizing allocation within the set of EFX allocations.

Envy-Free and Efficient Allocations for Graphical Valuations

TL;DR

It is shown that the price of EFX with respect to utilitarian welfare is one for binary utilities, but can be arbitrarily large {0, 1, d} valuations, but can be arbitrarily large {0, 1, d} valuations.

Abstract

We consider the complexity of finding envy-free allocations for the class of graphical valuations. Graphical valuations were introduced by Christodoulou et. al.(2023) as a structured class of valuations that admit allocations that are envy-free up to any item (EFX). These are valuations where every item is valued by two agents, lending a (simple) graph structure to the utilities, where the agents are vertices and are adjacent if and only if they value a (unique) common item. Finding envy-free allocations for general valuations is known to be computationally intractable even for very special cases: in particular, even for binary valuations, and even for identical valuations with two agents. We show that, for binary graphical valuations, the existence of envy-free allocations can be determined in polynomial time. In contrast, we also show that allowing for even slightly more general utilities {0, 1, d} leads to intractability even for graphical valuations. This motivates other approaches to tractability, and to that end, we exhibit the fixed-parameter tractability of the problem parameterized by the vertex cover number of the graph when the number of distinct utilities is bounded. We also show that, all graphical instances that admit EF allocations also admit one that is non-wasteful. Since EFX allocations are possibly wasteful, we also address the question of determining the price of fairness of EFX allocations. We show that the price of EFX with respect to utilitarian welfare is one for binary utilities, but can be arbitrarily large {0, 1, d} valuations. We also show the hardness of deciding the existence of an EFX allocation which is also welfare-maximizing and of finding a welfare-maximizing allocation within the set of EFX allocations.

Paper Structure

This paper contains 17 sections, 16 theorems, 7 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions.
  3. Additional Related Work.
  4. Preliminaries
  5. Model.
  6. Graphical Allocation Instance.
  7. Fairness Notions.
  8. Welfare Notions and Price of Fairness.
  9. Envy-Free Graphical Allocations
  10. Forward Direction.
  11. Reverse Direction.
  12. EFX and Welfare-Maximization
  13. Forward Direction.
  14. Reverse Direction.
  15. Forward Direction.
  16. ...and 2 more sections

Key Result

theorem thmcountertheorem

Given a graphical allocation instance, there is an EF allocation if and only if there is an EF orientation.

Figures (2)

  • Figure 1: A schematic of reduced instance in the proof of \ref{['thm:EFHard']}.
  • Figure 2: A schematic of reduced instance in the proof of \ref{['thm:utilhard']}.

Theorems & Definitions (27)

  • theorem thmcountertheorem
  • proof
  • lemma thmcounterlemma
  • proof
  • corollary thmcountercorollary
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • ...and 17 more