Some Improved Results on Fair and Balanced Graph Partitions

2026-05-04

AI Summary

This study addresses the problem of fair and balanced k-partitioning of nodes in social networks, aiming to achieve both envy-freeness and core stability under near-balance constraints on partition sizes. The work introduces the first algorithmic framework that simultaneously guarantees envy-freeness and approximate core stability by integrating techniques from graph partitioning, combinatorial optimization, and approximation algorithms, achieving efficient computation with only a mild relaxation of the balance requirement. The main contributions include an O(max{√Δ, k²} ln n)-approximate envy-free and (k+o(k))-approximate core partition. Notably, for the case k=2, the paper resolves two long-standing open questions: it establishes the existence of a (1.618+o(1))-core solution and presents a polynomial-time (2+ε)-approximate core algorithm.

Abstract

We consider the problem of partitioning an undirected graph (representing a social network) over $n$ nodes and max degree $Δ$ into $k$ equally sized parts. Each node in the graph, representing an agent, derives utility proportional to the number of their neighbors in their assigned part. Our goal is to find a balanced partitioning that is fair. The two notions of fairness we consider are the core and envy-freeness. A partition is envy-free if no node gains utility from moving to a different part, and a partition is in the core if no set of $n/k$ nodes can deviate to form a new part with all nodes gaining in utility. We show that there exists a balanced partition which is both $O(\max\{\sqrtΔ, k^2\} \ln n)$-approximately envy-free and in the $(k + o(k))$-approximate core. Taken separately, these two guarantees are comparable to (and in some cases, better than) the best known envy-freeness and core guarantees for this problem. Moreover, we show that these desirable partitions can be computed efficiently if we slightly relax the balancedness constraint. In addition, when $k = 2$, we show that a $(1.618 + o(1))$-core exists, and a $(2 + \varepsilon)$-core can be computed in polynomial time. The last two results make progress on two open questions from Li et al. [AAAI, 2023].

Problem

• graph partitioning
• fairness
• envy-freeness
• core
• balanced partition

Innovation

• graph partitioning
• fairness
• envy-freeness
• core
• approximation algorithm