Beyond Exact Fairness: Envy-Free Incomplete Connected Fair Division

TL;DR

This paper investigates the computational complexity of envy-free fair division with connected shares, proving hardness results for exact solutions and providing efficient approximation schemes when slight envy is tolerated.

Contribution

It establishes the hardness of exact envy-free connected fair division for certain parameters and introduces an efficient approximation scheme for near-envy-free solutions.

Findings

Exact problem is computationally hard for fixed parameters.

Approximation scheme efficiently finds near-envy-free solutions.

Results hold for general graphs and various input representations.

Abstract

We study the problem of Envy-Free Incomplete Connected Fair Division, where exactly p vertices of an undirected graph must be allocated to agents such that each agent receives a connected share and does not envy another agent's share. Focusing on agents with additive valuations, we show that the problem remains computationally hard when parameterized by p and the number of agents. This result holds even for star graphs and with the input numbers given in unary representation, thereby resolving an open problem posed by Gahlawat and Zehavi (FSTTCS 2023). In stark contrast, we show that if one is willing to tolerate even the slightest amount of envy, then the problem becomes efficient with respect to the natural parameters. Specifically, we design an Efficient Parameterized Approximation Scheme parameterized by p and the number of agent types. Our algorithm works on general graphs and remains efficient even when the input numbers are provided in binary representation.