Balanced and Fair Partitioning of Friends

TL;DR

The paper studies partitioning agents on a friendship graph into $k$ balanced groups with utilities driven by intra-group friendships, seeking fair allocations under multiple fairness notions. It generalizes prior binary/additive models to richer, non-binary utilities and adapts standard fair-division notions (EF, EFX, EF1, PROP, MMS) to this setting. The authors establish a comprehensive complexity landscape: in general graphs many fairness notions are NP-hard or W[1]-hard with respect to the number of parts, while positive results appear under structural restrictions such as small vertex cover or tree-like graphs; they provide polynomial-time algorithms for several cases (notably with binary utilities and bounded treewidth) and a dynamic programming framework over nice tree decompositions. They also connect fair partitioning to additively separable hedonic games with fixed-size coalitions (ASHG-FSC), demonstrating robustness of their methods under the broader model. The work highlights a rich interplay between graph structure, fairness concepts, and computational feasibility, offering both hardness results and tractable algorithms with practical impact for equitable clustering, team formation, and related clustering tasks. Applicability extends to ASHG-FSC, enabling fair allocations under fixed-size coalitions in broader strategic settings.” wrapped in $...$ where appropriate.

Abstract

In the recently introduced model of fair partitioning of friends, there is a set of agents located on the vertices of an underlying graph that indicates the friendships between the agents. The task is to partition the graph into $k$ balanced-sized groups, keeping in mind that the value of an agent for a group equals the number of edges they have in that group. The goal is to construct partitions that are "fair", i.e., no agent would like to replace an agent in a different group. We generalize the standard model by considering utilities for the agents that are beyond binary and additive. Having this as our foundation, our contribution is threefold (a) we adapt several fairness notions that have been developed in the fair division literature to our setting; (b) we give several existence guarantees supported by polynomial-time algorithms; (c) we initiate the study of the computational (and parameterized) complexity of the model and provide an almost complete landscape of the (in)tractability frontier for our fairness concepts.

Figures (6)

• Figure 1: An instance and a PROP partition which is not EF.
• Figure 2: A basic overview of our algorithmic and complexity results for unrestricted friendship graphs. An arrow from notion $A$ to $B$ denotes that if a partition is fair with respect to $A$, then it is also fair with respect to $B$. Here, $k$ is the number of parts, $n$ is the number of agents, and $\operatorname{vc}$ denotes the vertex cover number of the friendship graph $G$.
• Figure 3: An illustration of the construction used to prove \ref{['thm:EF:NPh']}. In this example, we assume an instance of Unary Bin Packing with $S=(2,5,3)$, $B=2$, and $c=5$. Next to each agent $a_i^j$, we depict (in blue) the value of $\operatorname{u}_\forall(a_i^j)$. With differently colored backgrounds, we highlight one possible balanced $2$-partition for this instance.
• Figure 4: An illustration of the construction used in \ref{['thm:EFX:NPh']}. For guard-agents, which are depicted in blue, we have $\operatorname{u}_\forall(g_1) = \operatorname{u}_\forall(g_2) = 1$, and $\operatorname{u}_\forall(S_1) = \operatorname{u}_\forall(S_2) = B$, where $B$ is half of the sum of all integers given in an original instance of the Equitable Partition problem. The element-agents are in one-to-one correspondence with integers of the Equitable Partition problem, which is captured in its objective utility.
• Figure 5: A basic overview of our complexity results for trees.

Theorems & Definitions (65)

• Definition 1: Vertex Cover Number
• Definition 2: Treewidth
• Definition 3: Nice Tree Decomposition
• Definition 4: EF
• Definition 5: EFX$_0$
• Definition 6: EFX
• Definition 7: EF1
• Definition 8: PROP
• Definition 9: MMS
• Proposition 1