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Fair Allocation with Binary Valuations for Mixed Divisible and Indivisible Goods

Yasushi Kawase, Koichi Nishimura, Hanna Sumita

TL;DR

This paper provides a polynomial-time algorithm for the case when all divisible goods are identical and homogeneous, and demonstrates that the problem is NP-hard in general.

Abstract

The fair allocation of mixed goods, consisting of both divisible and indivisible goods, has been a prominent topic of study in economics and computer science. We define an allocation as fair if its utility vector minimizes a symmetric strictly convex function. This fairness criterion includes standard ones such as maximum egalitarian social welfare and maximum Nash social welfare. We address the problem of minimizing a given symmetric strictly convex function when agents have binary valuations. If only divisible goods or only indivisible goods exist, the problem is known to be solvable in polynomial time. In this paper, firstly, we demonstrate that the problem is NP-hard even when all indivisible goods are identical. This NP-hardness is established even for maximizing egalitarian social welfare or Nash social welfare. Secondly, we provide a polynomial-time algorithm for the problem when all divisible goods are identical. To accomplish these, we exploit the proximity structure inherent in the problem. This provides theoretically important insights into the hybrid domain of convex optimization that incorporates both discrete and continuous aspects.

Fair Allocation with Binary Valuations for Mixed Divisible and Indivisible Goods

TL;DR

This paper provides a polynomial-time algorithm for the case when all divisible goods are identical and homogeneous, and demonstrates that the problem is NP-hard in general.

Abstract

The fair allocation of mixed goods, consisting of both divisible and indivisible goods, has been a prominent topic of study in economics and computer science. We define an allocation as fair if its utility vector minimizes a symmetric strictly convex function. This fairness criterion includes standard ones such as maximum egalitarian social welfare and maximum Nash social welfare. We address the problem of minimizing a given symmetric strictly convex function when agents have binary valuations. If only divisible goods or only indivisible goods exist, the problem is known to be solvable in polynomial time. In this paper, firstly, we demonstrate that the problem is NP-hard even when all indivisible goods are identical. This NP-hardness is established even for maximizing egalitarian social welfare or Nash social welfare. Secondly, we provide a polynomial-time algorithm for the problem when all divisible goods are identical. To accomplish these, we exploit the proximity structure inherent in the problem. This provides theoretically important insights into the hybrid domain of convex optimization that incorporates both discrete and continuous aspects.
Paper Structure (17 sections, 36 theorems, 40 equations, 1 figure, 4 tables, 1 algorithm)

This paper contains 17 sections, 36 theorems, 40 equations, 1 figure, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related work
  4. Organization
  5. Preliminaries
  6. Integral base-polyhedra and M-convex sets
  7. Definitions and properties
  8. The hybrid problem and relation to fair allocation
  9. Principal Partition and Canonical Partition
  10. Structure of Phi-minimizers
  11. The peak-set $N_1$
  12. Decomposition
  13. Structures in Fair Allocation
  14. Tractability for Identical Divisible Goods
  15. Hardness for Identical Indivisible Goods
  16. ...and 2 more sections

Key Result

Theorem 1.2

For any fixed symmetric strictly convex function $\Phi$, finding a $\Phi$-fair allocation is NP-hard even when indivisible goods are identical.

Figures (1)

  • Figure 1: The set of possible utility vectors in Example \ref{['ex:neither']}. The blue points are minimizers of $\Phi$.

Theorems & Definitions (57)

  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Fujishige fujishige1980 and Maruyama Maruyama1978
  • Theorem 2.2: Frank and Murota FM2022a
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Proposition 3.1: fujishige2005
  • ...and 47 more