Contiguous Allocation of Indivisible Items on a Path

2024-01-09

arXiv.org

AI Summary

This paper studies the contiguous allocation of indivisible goods on a path graph, where each agent must receive a connected subgraph and allocations must satisfy fairness (envy-freeness up to one good, EF1) and efficiency (utility maximization). Focusing on sparse preference structures—where each agent values at most $a$ goods and each good is valued by at most $b$ agents—the work establishes the first systematic computational complexity characterization. Methodologically, it employs hardness reductions, fixed-parameter tractable (FPT) algorithm design, and approximation techniques. Key contributions include: (i) proving NP-completeness of EF1 existence checking; (ii) developing an FPT algorithm parameterized by $a+b$ and a $1/a$-approximation algorithm for EF1; (iii) achieving FPT solvability and constant-factor approximations for both utilitarian and egalitarian social welfare; and (iv) showing polynomial-time exact solvability when the block order is pre-specified. The results yield a tight complexity classification and provide theoretically grounded, practical algorithms.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of allocating indivisible items on a path among agents. The objective is to find a fair and efficient allocation in which each agent's bundle forms a contiguous block on the line. We say that an instance is emph{$(a, b)$-sparse} if each agent values at most $a$ items positively and each item is valued positively by at most $b$ agents. We demonstrate that, even when the valuations are binary additive, deciding whether every item can be allocated to an agent who wants it is NP-complete for the $(4,3)$-sparse instances. Consequently, we provide two fixed-parameter tractable (FPT) algorithms for maximizing utilitarian social welfare, with respect to the number of agents and the number of items. Additionally, we present a $2$-approximation algorithm for the special case when the valuations are binary additive, and the maximum utility is equal to the number of items. Also, we provide a $1/a$-approximation algorithm for the $(a,b)$-sparse instances. Furthermore, we establish that deciding whether the maximum egalitarian social welfare is at least $2$ or at most $1$ is NP-complete for the $(6,3)$-sparse instances, even when the valuations are binary additive. We present a $1/a$-approximation algorithm for maximizing egalitarian social welfare for the $(a,b)$-sparse instances. Besides, we give two FPT algorithms for maximizing egalitarian social welfare in terms of the number of agents and the number of items. We also explore the case where the order of the blocks of items allocated to the agents is predetermined. In this case, we show that both maximum utilitarian social welfare and egalitarian social welfare can be computed in polynomial time. However, we determine that checking the existence of an EF1 allocation is NP-complete, even when the valuations are binary additive.

Problem

• Resource Allocation
• Fairness
• Approximation Algorithms

Innovation

• Indivisible Items Allocation
• Equitable Distribution Strategies
• Half-Optimal Solution Approaches