Parameterized Complexity of Incomplete Connected Fair Division

TL;DR

This paper studies the computational complexity of fair division on graphs, introducing a generalized problem where only a subset of items is allocated, and analyzes the difficulty of achieving various fairness criteria.

Contribution

It introduces Incomplete Connected Fair Division, analyzes its complexity under different fairness notions, and provides algorithms and kernelization results for these problems.

Findings

EF, EFO, and EFX fairness are W[1]-hard parameterized by p and number of agents.

A randomized FPT algorithm exists for PROP fairness parameterized by p.

Complexity results vary with parameters p, vertex cover number, and valuation diversity.

Abstract

\textit{Fair division} of resources among competing agents is a fundamental problem in computational social choice and economic game theory. It has been intensively studied on various kinds of items (\textit{divisible} and \textit{indivisible}) and under various notions of \textit{fairness}. We focus on Connected Fair Division (\CFDO), the variant of fair division on graphs, where the \textit{resources} are modeled as an \textit{item graph}. Here, each agent has to be assigned a connected subgraph of the item graph, and each item has to be assigned to some agent. We introduce a generalization of \CFDO, termed Incomplete Connected Fair Division (\CFD), where exactly $p$ vertices of the item graph should be assigned to the agents. This might be useful, in particular when the allocations are intended to be ``economical'' as well as fair. We consider four well-known notions of fairness: \PROP, \EF, \EFO, \EFX. First, we prove that \EF-\CFD, \EFO-\CFD, and \EFX-\CFD are W[1]-hard parameterized by $p$ plus the number of agents, even for graphs having constant \textit{vertex cover number} ($\mathsf{vcn}$). In contrast, we present a randomized \FPT algorithm for \PROP-\CFD parameterized only by $p$. Additionally, we prove both positive and negative results concerning the kernelization complexity of \CFD under all four fairness notions, parameterized by $p$, $\mathsf{vcn}$, and the total number of different valuations in the item graph ($\mathsf{val}$).