Inner-group Maximization (IGM) Overview

• IGM is a multi-disciplinary concept defining distinct intra-group and agent-level optimization strategies in fair influence, MARL, and graph centrality.
• In fair influence maximization, IGM isolates submodular per-group influence functions to enable a two-step decomposition for efficient seed selection.
• In cooperative MARL and graph optimization, IGM ensures decentralized actions align with joint value functions and guides group-centric subset selection.

Searching arXiv for the cited papers to ground the article in current records. arXiv search query: (Rui et al., 30 Sep 2025) Efficient Approximation Algorithms for Fair Influence Maximization under Maximin Constraint Inner-group Maximization (IGM) is a polysemous research term whose meaning depends on disciplinary context. In fair influence maximization, it denotes the first stage of a two-step optimization framework that exploits submodularity within demographic or community partitions to generate high-quality seed lists for each group (Rui et al., 30 Sep 2025). In cooperative multi-agent reinforcement learning (MARL), the same acronym more commonly denotes Individual Global Max, a structural condition requiring decentralized greedy actions to coincide with the argmax of a joint value function (Xu et al., 2023, Hong et al., 2022). In graph optimization, “inner-group maximization” also describes problems such as selecting a size-$k$ subset that optimizes a group-level centrality objective, as in group closeness maximization (Bergamini et al., 2017). The term therefore refers not to a single formal object but to a family of intra-group or per-agent optimization constructs whose algorithmic significance is tied to decomposition, structural properties, and the tractability of otherwise difficult global objectives.

1. Terminological scope and field-specific meanings

In the cited literature, IGM denotes distinct concepts rather than a single canonical definition. The principal usages are summarized below.

Usage of IGM	Domain	Core meaning
Inner-group Maximization	Fair influence maximization	Per-group optimization of $\sigma_c(S)$ before across-group coordination
Individual Global Max	Cooperative MARL	Equality between global joint argmax and collection of local argmaxes
Inner-group maximization problem	Graph centrality / facility-location-like optimization	Selection of a size-$k$ subset optimizing a group objective

The fair influence maximization usage is explicitly structural: the global fairness objective is non-submodular, but the per-group influence function remains submodular, so optimization is decomposed into Inner-group Maximization (IGM) and Across-group Maximization (AGM) (Rui et al., 30 Sep 2025). In MARL, by contrast, IGM is a consistency condition linking $Q_{\mathrm{tot}}(s,\boldsymbol{u})$ and per-agent utilities $Q_a(\tau^a,u^a)$, and it underlies CTDE value decomposition methods such as VDN, QMIX, QTRAN, QPLEX, Weighted QMIX, and MAVEN (Xu et al., 2023, Hong et al., 2022). In network centrality, Bergamini et al. describe Group Closeness Maximization as an inner-group maximization problem because it selects a subset $S\subseteq V$ of prescribed size $k$ that optimizes a group-level objective (Bergamini et al., 2017).

This terminological overloading is a recurrent source of confusion. A plausible implication is that any technical discussion of “IGM” should first specify whether it concerns groupwise combinatorial optimization, decentralized argmax consistency in MARL, or a more general subset-selection problem.

2. Inner-group Maximization in fair influence maximization

In "Efficient Approximation Algorithms for Fair Influence Maximization under Maximin Constraint" (Rui et al., 30 Sep 2025), the social network is modeled as a directed graph $G=(V,E)$, with the population partitioned into $m$ non-overlapping groups $C=\{c_1,c_2,\dots,c_m\}$ and $\sigma_c(S)$0. Diffusion follows the Independent Cascade model, with edge activation probability $\sigma_c(S)$1, where $\sigma_c(S)$2 is the in-degree of $\sigma_c(S)$3. For a seed set $\sigma_c(S)$4, total expected spread is denoted $\sigma_c(S)$5, while inner-group influence for group $\sigma_c(S)$6 is

$\sigma_c(S)$7

The group utility is the influenced ratio

$\sigma_c(S)$8

and the fairness objective is the maximin criterion

$\sigma_c(S)$9

The paper formulates Fair Influence Maximization under a global seed budget $k$0 as finding

$k$1

The central difficulty is that $k$2 is non-submodular, so standard submodular maximization tools do not directly apply. The proposed response is a two-step decomposition. First, Inner-group Maximization independently computes, for each group $k$3, an ordered seed list

$k$4

that is good for maximizing $k$5. Second, Across-group Maximization coordinates selections across the per-group lists to build the final global seed set.

The IGM subproblem for a fixed group $k$6 is

$k$7

with optimum

$k$8

A notable modeling choice is that seeds are selected from the entire network rather than only from $k$9. Thus, a seed outside $Q_{\mathrm{tot}}(s,\boldsymbol{u})$0 may be chosen if it exerts strong influence on $Q_{\mathrm{tot}}(s,\boldsymbol{u})$1 via cross-group paths. The output of IGM over all groups is

$Q_{\mathrm{tot}}(s,\boldsymbol{u})$2

This decomposition isolates the subproblem that remains amenable to efficient approximation. The paper’s explicit rationale is that IGM solves $Q_{\mathrm{tot}}(s,\boldsymbol{u})$3 independent influence maximization problems, whereas AGM enforces the global fairness constraint over a reduced candidate space.

3. Structural properties, algorithmic realization, and guarantees

The foundational result for fair-influence IGM is the lemma that, for each group $Q_{\mathrm{tot}}(s,\boldsymbol{u})$4, the inner influence spread $Q_{\mathrm{tot}}(s,\boldsymbol{u})$5 is submodular under the IC model (Rui et al., 30 Sep 2025). Using the live-edge graph view, the paper defines $Q_{\mathrm{tot}}(s,\boldsymbol{u})$6 as the nodes reachable from $Q_{\mathrm{tot}}(s,\boldsymbol{u})$7 in a sampled live-edge graph $Q_{\mathrm{tot}}(s,\boldsymbol{u})$8, then $Q_{\mathrm{tot}}(s,\boldsymbol{u})$9, and writes

$Q_a(\tau^a,u^a)$0

For fixed $Q_a(\tau^a,u^a)$1, $Q_a(\tau^a,u^a)$2 is a reachability function with diminishing returns; expectation over $Q_a(\tau^a,u^a)$3 preserves submodularity. The same section states that $Q_a(\tau^a,u^a)$4 is also monotone and non-negative.

By contrast, the global objective

$Q_a(\tau^a,u^a)$5

is the minimum of submodular functions scaled by positive constants, and the minimum of submodular functions is not generally submodular. The paper’s intuition is that the identity of the worst-off group can switch as seeds are added, breaking diminishing returns.

Given monotone submodularity and the cardinality constraint $Q_a(\tau^a,u^a)$6, the classical greedy principle yields a $Q_a(\tau^a,u^a)$7-approximation for maximizing $Q_a(\tau^a,u^a)$8. Exact evaluation is $Q_a(\tau^a,u^a)$9-hard, so the implementation uses IMM, adapted to the inner-group objective. For group $S\subseteq V$0, IMM generates

$S\subseteq V$1

RR sets, where $S\subseteq V$2, $S\subseteq V$3 is a confidence parameter, $S\subseteq V$4 is the optimal inner-group influence, and $S\subseteq V$5 is an error parameter. Greedy coverage of these RR sets yields, with probability at least $S\subseteq V$6,

$S\subseteq V$7

IGM stores the resulting ordered seed lists and the RR sets generated in each per-group IMM run. These RR sets are reused in AGM to evaluate $S\subseteq V$8 efficiently for arbitrary candidate seed sets. The paper emphasizes that IGM does not enforce the global budget across groups; each group is optimized independently with budget $S\subseteq V$9, and the actual global budget is enforced only in AGM.

AGM introduces two coordination strategies. AGM-US selects approximately uniformly across columns of the seed matrix and guarantees

$k$0

AGM-GS greedily selects from the current frontier candidates across groups and, when groups are completely disconnected, guarantees

$k$1

The paper also states that AGM-GS may fail to preserve this guarantee in general graphs with cross-group edges, although empirically it often outperforms AGM-US.

The computational motivation for IGM is equally explicit. Running IMM once per group leads to complexity roughly $k$2, and the method inherits the scalability benefits of Reverse Influence Sampling. Because per-group IMM runs are independent, the procedure is embarrassingly parallel. This suggests that the decomposition is not merely analytical; it is also a practical mechanism for turning a non-submodular fairness objective into a scalable pipeline.

4. Individual Global Max in cooperative MARL

In cooperative MARL, IGM usually denotes Individual Global Max rather than Inner-group Maximization. The defining statement, used in "Dual Self-Awareness Value Decomposition Framework without Individual Global Max for Cooperative Multi-Agent Reinforcement Learning" and in "Rethinking Individual Global Max in Cooperative Multi-Agent Reinforcement Learning," is that the optimal joint action should be obtainable by combining individually optimal local actions (Xu et al., 2023, Hong et al., 2022).

The setting is a fully cooperative Dec-POMDP or DEC-POMDP with agents $k$3, global state $k$4, joint action $k$5, and joint action-value

$k$6

Value decomposition methods introduce per-agent utilities $k$7, where $k$8 is agent $k$9’s action-observation history, together with a mixing function that reconstructs $G=(V,E)$0.

The IGM condition is stated as

$G=(V,E)$1

and also, in the notation of the same literature,

$G=(V,E)$2

Operationally, decentralized execution then takes the form

$G=(V,E)$3

with the requirement that the resulting joint action be globally optimal.

This principle organizes several major value decomposition methods. VDN uses additive factorization,

$G=(V,E)$4

QMIX uses a monotonic mixing network with the structural constraint

$G=(V,E)$5

which guarantees compatibility with IGM. QTRAN drops structural monotonicity but enforces IGM through constraints or regularization. QPLEX, Weighted QMIX, and related variants continue to revolve around IGM or its variants.

Within CTDE, IGM is therefore both a decentralization condition and a representational restriction. It ensures that greedy local decisions can recover the joint argmax without centralized coordination at execution time.

5. Critiques of IGM and IGM-free alternatives in MARL

Two distinct critiques appear in the cited MARL literature. The first is representational: IGM-based factorization can be too restrictive for non-monotonic joint value landscapes. The second is informational: under partial observability, the decomposition from global state to local observation is inherently lossy, and embedding that decomposition inside Bellman iterations causes error accumulation (Xu et al., 2023, Hong et al., 2022).

The representational critique is developed in DAVE. The paper argues that non-negative-weight and monotonicity constraints restrict the set of representable joint value functions. Its synthetic single-state example uses two agents with 101 actions each, local utilities

$G=(V,E)$6

and joint reward

$G=(V,E)$7

An IGM-based QMIX mixer with non-negative weights cannot accurately fit this non-monotonic structure, whereas an unrestricted IGM-free mixer fits it much better and faster. The same paper reports that DAVE is the first fully IGM-free value decomposition method and replaces IGM with a dual self-awareness architecture: each agent has an alter ego value function $G=(V,E)$8, mixed by an unrestricted IGM-free network into $G=(V,E)$9, and an ego policy $m$0 used for action selection. Instead of requiring local argmaxes to match the global argmax, DAVE samples a set of joint actions

$m$1

then chooses

$m$2

Its alter ego is trained with an Expected SARSA target, and its ego policy is trained by supervised learning. The paper adds an anti-ego exploration mechanism based on an auto-encoder novelty score

$m$3

The combined ego loss is

$m$4

with $m$5 annealed linearly. Empirically, the paper reports that $m$6 fails, $m$7 already yields strong performance, and larger $m$8 gives diminishing returns; it also states that DAVE is the only method that reliably learns the optimal policy in the multi-step matrix game and that, in 6h_vs_8z, DAVE is the only method with non-zero median win ratio.

The informational critique is developed in IGM-DA. The paper defines insufficient observation as the case where there exist $m$9 such that $C=\{c_1,c_2,\dots,c_m\}$0, and defines lossy decomposition as the existence of $C=\{c_1,c_2,\dots,c_m\}$1 and $C=\{c_1,c_2,\dots,c_m\}$2 such that

$C=\{c_1,c_2,\dots,c_m\}$3

It proves that if $C=\{c_1,c_2,\dots,c_m\}$4 is an insufficient observation of $C=\{c_1,c_2,\dots,c_m\}$5, then there exists some joint value function for which decomposition to local utilities is lossy. When hypernetwork-based IGM decomposition is inserted directly into Bellman iterations, the resulting error is

$C=\{c_1,c_2,\dots,c_m\}$6

so lossy decomposition error accumulates over time. IGM-DA separates these roles by training global-state expert networks $C=\{c_1,c_2,\dots,c_m\}$7 and a global-state IGM mixer for RL, then training local-observation learners $C=\{c_1,c_2,\dots,c_m\}$8 by imitation. Under this design,

$C=\{c_1,c_2,\dots,c_m\}$9

so decomposition error is not accumulated through the Bellman recursion. The imitation stage uses

$\sigma_c(S)$00

with expert policy

$\sigma_c(S)$01

The paper recommends DAgger rather than Behavioral Cloning because DAgger aligns expert labels with the learner’s visitation distribution. On SMAC with sight-view $\sigma_c(S)$02, the reported average win rates are 47.8 for QMIX versus 68.9 for QMIX-DA, 41.0 for QPLEX versus 63.1 for QPLEX-DA, and 50.4 for DMIX versus 67.0 for DMIX-DA.

A common misconception in this literature is that IGM is merely a convenient engineering convention. The cited papers instead treat it as a deep structural assumption: DAVE rejects it entirely, while IGM-DA retains it only for global-state experts and removes it from the local Bellman loop.

6. Group closeness maximization as an inner-group optimization problem

"Scaling up Group Closeness Maximization" treats the selection of a size-$\sigma_c(S)$03 subset optimizing a group centrality as an inner-group maximization problem (Bergamini et al., 2017). For a graph $\sigma_c(S)$04 and shortest-path distance $\sigma_c(S)$05, the distance from a node $\sigma_c(S)$06 to a group $\sigma_c(S)$07 is

$\sigma_c(S)$08

Using the formulation of Bergamini et al., group closeness is

$\sigma_c(S)$09

and group farness is

$\sigma_c(S)$10

The Group Closeness Maximization problem is

$\sigma_c(S)$11

A central theoretical correction in the paper is that group closeness is not submodular, even though group farness is supermodular. Bergamini et al. give a counterexample on the complete graph $\sigma_c(S)$12 with $\sigma_c(S)$13, $\sigma_c(S)$14, and $\sigma_c(S)$15, obtaining

$\sigma_c(S)$16

Hence

$\sigma_c(S)$17

but

$\sigma_c(S)$18

which violates diminishing returns. The paper therefore states that the $\sigma_c(S)$19 approximation factor previously claimed for greedy group closeness does not hold.

The constructive part of the paper is algorithmic rather than approximation-theoretic. Greedy++ combines a memory-efficient greedy structure, pruned SSSP, and Minoux-style lazy evaluation. The method maintains a single distance vector $\sigma_c(S)$20, evaluates candidates using

$\sigma_c(S)$21

and prunes SSSP when a visited node $\sigma_c(S)$22 satisfies $\sigma_c(S)$23, because descendants cannot improve the objective. Supermodularity of farness implies that marginal farness changes

$\sigma_c(S)$24

are non-decreasing as the group grows, enabling lazy skipping of many candidate evaluations.

For unweighted graphs, the paper also gives a bit-parallel formulation: $\sigma_c(S)$25 This accelerates marginal-gain computation through bitwise operations but requires $\sigma_c(S)$26 bits of storage.

The empirical scalability claims are specific. Greedy++ is reported to handle networks with up to hundreds of millions of edges in minutes or at most a few hours; on com-orkut, the reported sequential time for $\sigma_c(S)$27 is about 60,252 seconds and the parallel time with 16 threads is 5,793 seconds; on youtube-u-growth, the corresponding times are about 22,299 seconds sequential and about 2,196 seconds parallel. On instances where the optimum can be computed by ILP, Greedy++ achieves empirical approximation ratios of 0.998 for $\sigma_c(S)$28, 0.994 for $\sigma_c(S)$29, and 0.995 for $\sigma_c(S)$30. The paper therefore treats Greedy++ as a heuristic without a worst-case constant-factor guarantee, but one that is extremely close to optimal in experiments.

Across this literature, the group closeness case provides a useful contrast to fair influence maximization. In both settings, the global objective resists straightforward submodular optimization. In fair influence maximization, the remedy is decomposition into submodular per-group influence objectives; in group closeness, the remedy is to work with the supermodularity of a dual objective, farness, to accelerate greedy search. This suggests that “inner-group maximization” is best understood as a methodological pattern: isolate a structurally exploitable intra-group or per-component objective, then coordinate or evaluate globally only after the expensive combinatorial core has been reduced.