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The Complexity of Extending Fair Allocations of Indivisible Goods

Argyrios Deligkas, Eduard Eiben, Robert Ganian, Tiger-Lily Goldsmith, Stavros D. Ioannidis

TL;DR

We study the envy-free allocation extension problem for indivisible goods with additive valuations, where a fixed partial allocation must be extended to the remaining items. The authors provide a comprehensive complexity classification under natural parameters, including W[1]-hardness for the number of open items $k$ (even when considering item-types) and fixed-parameter tractability when combining $k$ with the number of agent types $n_t$, plus a detailed map for settings with limited item types $m_t$. A central technical contribution is a reduction from Multicolored Clique establishing the $W[1]$-hardness, complemented by algorithmic approaches (e.g., partitioning by agent types and matching) that yield FPT results for certain parameter pairs. They further analyze extensions beyond envy-freeness, showing EF1 can always extend an envy-free partial allocation while noting inherent limitations for extending EFX, and they discuss implications for related fairness notions and practical allocation scenarios.

Abstract

We initiate the study of computing envy-free allocations of indivisible items in the extension setting, i.e., when some part of the allocation is fixed and the task is to allocate the remaining items. Given the known NP-hardness of the problem, we investigate whether -- and under which conditions -- one can obtain fixed-parameter algorithms for computing a solution in settings where most of the allocation is already fixed. Our results provide a broad complexity-theoretic classification of the problem which includes: (a) fixed-parameter algorithms tailored to settings with few distinct types of agents or items; (b) lower bounds which exclude the generalization of these positive results to more general settings. We conclude by showing that -- unlike when computing allocations from scratch -- the non-algorithmic question of whether more relaxed EFX allocations exist can be completely resolved in the extension setting.

The Complexity of Extending Fair Allocations of Indivisible Goods

TL;DR

We study the envy-free allocation extension problem for indivisible goods with additive valuations, where a fixed partial allocation must be extended to the remaining items. The authors provide a comprehensive complexity classification under natural parameters, including W[1]-hardness for the number of open items (even when considering item-types) and fixed-parameter tractability when combining with the number of agent types , plus a detailed map for settings with limited item types . A central technical contribution is a reduction from Multicolored Clique establishing the -hardness, complemented by algorithmic approaches (e.g., partitioning by agent types and matching) that yield FPT results for certain parameter pairs. They further analyze extensions beyond envy-freeness, showing EF1 can always extend an envy-free partial allocation while noting inherent limitations for extending EFX, and they discuss implications for related fairness notions and practical allocation scenarios.

Abstract

We initiate the study of computing envy-free allocations of indivisible items in the extension setting, i.e., when some part of the allocation is fixed and the task is to allocate the remaining items. Given the known NP-hardness of the problem, we investigate whether -- and under which conditions -- one can obtain fixed-parameter algorithms for computing a solution in settings where most of the allocation is already fixed. Our results provide a broad complexity-theoretic classification of the problem which includes: (a) fixed-parameter algorithms tailored to settings with few distinct types of agents or items; (b) lower bounds which exclude the generalization of these positive results to more general settings. We conclude by showing that -- unlike when computing allocations from scratch -- the non-algorithmic question of whether more relaxed EFX allocations exist can be completely resolved in the extension setting.

Paper Structure

This paper contains 19 sections, 15 theorems, 5 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Preliminaries
  4. Partial and Extended Allocations.
  5. Fairness Concepts.
  6. Fair Allocation Extension Problems.
  7. Parameterized complexity.
  8. Parameterizing by the Number of Open Items
  9. Proof of Theorem \ref{['thm:kmtwhard']}
  10. Construction.
  11. Partial allocation.
  12. From $\mathcal{I}$ to $\mathcal{I}'$.
  13. From $\mathcal{I}'$ to $\mathcal{I}$.
  14. Parameterizing by the Number of Recipients
  15. Construction.
  16. ...and 4 more sections

Key Result

Theorem 1

Envy-Free Allocation Extension is fixed-parameter tractable when parameterized by the number $k$ of open items plus the number $n_t$ of agent types.

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 5
  • proof
  • Theorem 5
  • Definition 1: Adjacent agents
  • Lemma 1
  • ...and 14 more