Efficient Approximation Algorithms for Fair Influence Maximization under Maximin Constraint

Fair Influence Maximization (FIM) seeks to mitigate disparities in influence across different groups and has recently garnered increasing attention. A widely adopted notion of fairness in FIM is the maximin constraint, which directly requires maximizing the utility (influenced ratio within a group) of the worst-off group. Despite its intuitive formulation, designing efficient algorithms with strong theoretical guarantees remains challenging, as the maximin objective does not satisfy submodularity, a key property for designing approximate algorithms in traditional influence maximization settings. In this paper, we address this challenge by proposing a two-step optimization framework consisting of Inner-group Maximization (IGM) and Across-group Maximization (AGM). We first prove that the influence spread within any individual group remains submodular, enabling effective optimization within groups. Based on this, IGM applies a greedy approach to pick high-quality seeds for each group. In the second step, AGM coordinates seed selection across groups by introducing two strategies: Uniform Selection (US) and Greedy Selection (GS). We prove that AGM-GS holds a $(1-1/e-\varepsilon)$ approximation to the optimal solution when groups are completely disconnected, while AGM-US guarantees a roughly $\frac{1}{m}(1-1/e-\varepsilon)$ lower bound regardless of the group structure, with $m$ denoting the number of groups.

💡 Research Summary

The paper tackles the Fair Influence Maximization (FIM) problem under the maximin fairness constraint, which seeks to maximize the influence ratio of the worst‑off demographic group. While the classic influence maximization objective is submodular and admits a (1‑1/e) greedy approximation, the maximin objective is not submodular, making it difficult to design efficient algorithms with provable guarantees. To overcome this, the authors propose a two‑step optimization framework consisting of Inner‑group Maximization (IGM) and Across‑group Maximization (AGM).

In the IGM phase, the authors prove that the influence spread restricted to any single group, denoted σ_c(S), remains a monotone submodular function under the Independent Cascade (IC) model. This property enables the use of the well‑known IMM (Influence Maximization via Martingales) algorithm, which is based on Reverse Influence Sampling (RIS), to obtain a (1‑1/e‑ε) approximation of the optimal seed set for each group independently. Importantly, the seeds selected for a group are not forced to lie inside the group, preserving the global search space.

The AGM phase coordinates the seed sets produced by IGM to satisfy the overall budget k. Two coordination strategies are introduced:

1. Uniform Selection (AGM‑US) – seeds are taken column‑wise from the m groups’ seed lists. When the remaining budget is insufficient to take an entire column, the algorithm greedily picks the node that yields the largest increase in the maximin objective Φ(S). The authors prove that AGM‑US guarantees Φ(S) ≥ (1/m − ξ)(1‑1/e‑ε)·Φ(S*), where ξ = mod(k,m)/k and S* is the optimal solution. This bound holds for any group structure, and in practice the bound can be tighter because duplicate nodes across groups reduce the effective column size.

2. Greedy Selection (AGM‑GS) – the algorithm repeatedly selects the node that maximally improves Φ(S) from the union of all candidate seeds. While a general approximation guarantee is elusive, the authors show that when groups are completely disconnected (no cross‑group edges), AGM‑GS achieves the same (1‑1/e‑ε) approximation as the classic submodular case. Empirically, AGM‑GS consistently outperforms AGM‑US even on graphs with cross‑group edges.

Complexity analysis reveals that IGM runs in O((|V_c|+|E_c|)·(k/ε²·log|V_c|)) per group, AGM‑US runs in O(k·m), and AGM‑GS runs in O(k·|V|). Hence the entire pipeline scales to networks with tens of thousands of nodes.

Extensive experiments on seven real‑world datasets (social media, citation networks, etc.) with varying numbers of groups, group size imbalance, and inter‑group edge density validate the theoretical claims. The results show: (i) AGM‑GS attains maximin utilities close to the optimal, (ii) AGM‑US meets its theoretical lower bound but suffers more when inter‑group connectivity is high, (iii) both methods are orders of magnitude faster than prior MILP or LP‑Greedy approaches, and (iv) the price of fairness (loss in total spread) is modest, especially for AGM‑GS.

In summary, the paper delivers a principled, scalable solution to maximin‑fair influence maximization by exploiting submodularity at the group level and providing two practical coordination mechanisms with provable guarantees. The work bridges the gap between fairness‑oriented objectives and algorithmic efficiency, opening avenues for future extensions such as tighter guarantees for AGM‑GS on general graphs, dynamic networks, and multi‑objective formulations that jointly optimize total spread and fairness.