Min-Cost Max-Influence Subgraphs

• Minimum-cost maximum-influence subgraph generation is a computational challenge of constructing subgraphs that connect specified terminals while optimizing influence spread within a given budget.
• It leverages greedy heuristics, integer programming, and polyhedral methods to balance cost and influence, achieving provable approximations and improved performance in complex networks.
• Experimental results demonstrate significant cost reductions and enhanced diffusion effectiveness, making it pivotal for applications in social influence and network information retrieval.

Minimum-cost maximum-influence subgraph generation refers to the computational problem of constructing, within a given budget, a subgraph of a larger network that connects specified terminal nodes, maximizing the aggregated influence over chosen nodes while minimizing the total cost incurred. This paradigm is central to problems in network science, information retrieval, and social influence maximization, with objectives and variants shaped by underlying models of diffusion, cost attribution, and influence quantification (Wang et al., 2 Nov 2025, Nguyen et al., 2012, Chen et al., 2022, Cordasco et al., 2015). The problem is fundamentally constrained by combinatorial complexity, reducing to classical hard graph problems in special cases, but novel algorithmic and polyhedral approaches provide theoretical and practical advances for real-world networks.

1. Formal Definitions and Problem Variants

Formally, minimum-cost maximum-influence subgraph generation seeks, for a host graph $G = (V, E)$:

• An influence score $\phi: V \to \mathbb{R}_+$ (node-level);
• An edge cost $c: E \to \mathbb{R}_+$;
• A set of terminal nodes $F \subseteq V$;
• Optionally, a propagation or activation model (e.g., Independent Cascade, Linear Threshold).

Given budget $C$, the task is to find a connected subgraph $S = (V_S, E_S)$ containing $F$, $\sum_{e \in E_S} c(e) \leq C$, that maximizes the sum of node influences $\sum_{v \in V_S} \phi(v)$ or the average $\sum_{v \in V_S} \phi(v)/\sum_{e \in E_S} c(e)$. In diffusion-based variants, the optimization is with respect to the expected spread of influence under the model, subject to budgeted seeding costs (Wang et al., 2 Nov 2025, Nguyen et al., 2012).

Specialized variants include:

• Budgeted Influence Maximization (BIM): maximize diffusion spread with node seeding costs and budget under probabilistic diffusion (Nguyen et al., 2012).
• Weighted Target Set Selection and Partial Incentivization: minimize initial seeding or incentive cost to ensure entire-network or high-coverage activation (Cordasco et al., 2015, Chen et al., 2022).

2. Computational Complexity

Minimum-cost maximum-influence subgraph generation is NP-hard across most variants:

• General Decision Problem: Determining if there exists a subgraph covering terminals $\phi: V \to \mathbb{R}_+$0 with influence at least $\phi: V \to \mathbb{R}_+$1 and total cost at most $\phi: V \to \mathbb{R}_+$2 is NP-hard via reduction from the Steiner tree problem (Wang et al., 2 Nov 2025).
• Diffusion Models: Computing expected spread (e.g., $\phi: V \to \mathbb{R}_+$3 under the IC model) is #P-complete even for DAGs (Nguyen et al., 2012).
• Seeding/Target Set Problems: For threshold-based activation, target set selection is hard to approximate within sub-exponential factors, unless NP ⊆ DTIME($\phi: V \to \mathbb{R}_+$4) (Cordasco et al., 2015).

This motivates both approximate algorithms (usually with provable constant-factor guarantees) and exact algorithms leveraging problem structure in restricted topologies (Chen et al., 2022).

3. Algorithmic Methods and Polyhedral Approaches

Several algorithmic frameworks address the problem:

3.1. Greedy Approximations and Heuristic Algorithms

• Steiner-Bicriteria Greedy (AGRAG/MCMI): Build a minimum-cost Steiner tree to cover $\phi: V \to \mathbb{R}_+$5, then repeatedly expand the subgraph by adding frontier vertices with the highest influence-per-unit-cost ratio if this improves the average, until no further improvement is possible. Achieves a 2-approximation to the cost relative to optimal (Wang et al., 2 Nov 2025).
• Ratio-based Node Selection (BIM): Iteratively choose nodes maximizing marginal influence gain per unit cost ($\phi: V \to \mathbb{R}_+$6), keeping within the budget. For classic models with submodular, monotone influence, achieves $\phi: V \to \mathbb{R}_+$7-approximation of optimal spread (Nguyen et al., 2012).
• Greedy-Peeling (Partial Incentive Models): Iteratively select vertices or adjust incentives based on local thresholds and degree; for some classes (trees, cliques) yields exact solutions; in general provides upper bounds on total cost in terms of node degrees and thresholds (Cordasco et al., 2015).

3.2. Polyhedral and Exact Methods

• Integer Programming (IP) Formulations: Minimize sum of incentive variables subject to activation constraints, propagation logic, and coverage requirements. Introduces variables for incentives, direction of influence ($\phi: V \to \mathbb{R}_+$8), and activated state ($\phi: V \to \mathbb{R}_+$9) (Chen et al., 2022).
• Cut-Generating and Facet-Defining Inequalities: Incorporate mixed 0–1 knapsack subcomponents, cycle-dominating inequalities, and minimum influencing subset (MIS) inequalities to strengthen the IP relaxation. Efficient separation algorithms for these inequalities reduce integrality gaps and accelerate convergence (Chen et al., 2022).
• Dynamic Programming on Simple Structures: For cycles and trees, specialized DP enables exact computation of minimum-cost incentive-appeal strategies activating all or a given fraction of nodes (Chen et al., 2022).

4. Influence and Cost Metrics

• Influence Score ($c: E \to \mathbb{R}_+$0): For retrieval-augmented tasks, node influence is often defined as Personalized PageRank with restarts at query terminals; in classical social-influence models, influence is the expected number of activated nodes or a linear combination based on weights (Wang et al., 2 Nov 2025, Nguyen et al., 2012, Cordasco et al., 2015).
• Edge/Node Costs: Edge costs may be semantic (e.g., similarity to a query for textual knowledge graphs), whereas node seeding costs typically measure intervention or incentive expense (Wang et al., 2 Nov 2025, Nguyen et al., 2012, Cordasco et al., 2015). Costs directly impact the feasible set of subgraphs and thus the tradeoff surface between influence and cost.

5. Theoretical Guarantees and Complexity

A spectrum of guarantees and complexity insights underpin the tractability and solution quality of proposed approaches:

Problem Variant	Guarantee	Complexity
Steiner-based MCMI	2-approximation in cost	$c: E \to \mathbb{R}_+$1 per (Wang et al., 2 Nov 2025)
BIM with submodular spread	$c: E \to \mathbb{R}_+$2-approx	$c: E \to \mathbb{R}_+$3 (Nguyen et al., 2012)
Partial Incentive (TPI/WTSS)	Non-constant upper bound	$c: E \to \mathbb{R}_+$4 (Cordasco et al., 2015)
Polyhedral (IP, cuts)	Exact for trees, strong LP	Polynomial in small graphs; $c: E \to \mathbb{R}_+$5 for complete separation (Chen et al., 2022)

The strongest constant-factor guarantees exist when both cost and influence are submodular/monotone, as in classic diffusion models. Otherwise, bicriteria or data-dependent upper bounds prevail.

6. Practical Implementations and Experimental Insights

Experimental validation spans retrieval-augmented generation with LLMs, social/diffusion benchmarks, and random graphs:

• AGRAG/MCMI for Retrieval-Augmented Generation: The MCMI subgraph gives explicit, multi-hop, and cyclic reasoning paths, guiding LLMs to more focused and faithful responses. Empirically, up to 3–6 point improvements in coverage and faithfulness, and up to 1.66× faster runtime versus state-of-the-art retrieval baselines (Wang et al., 2 Nov 2025).
• Budgeted Influence Maximization: Approximative greedy methods (DAG reductions + BP inference) yield influence spreads within 5–10% of MC-simulation-based greedy while being 2–3 orders of magnitude faster. On dense or flat-degree networks, advanced greedy outperforms simple heuristics by the largest margins (Nguyen et al., 2012).
• Polyhedral Algorithms in Partial Incentivization: Delayed cut generation with enriched cuts closes more instances to optimality and reduces root-LP primal-dual gaps by an order of magnitude compared to baseline branch-and-cut, especially as coverage rates and network density increase (Chen et al., 2022).
• Partial Incentive Greedy: In real networks, partial incentive strategies can yield 3–40× cost reduction over degree-based heuristics for activating the entire network (Cordasco et al., 2015).

7. Extensions, Relationships, and Significance

Minimum-cost maximum-influence subgraph generation unifies classical combinatorial optimization (e.g., Steiner tree, knapsack, submodular maximization) with contemporary diffusion, incentivization, and retrieval tasks in networks and knowledge graphs. Variants extend naturally to group-influence targets, dynamic incentives, and cost-influence tradeoff surfaces. Advances in polyhedral theory, approximation algorithms, and empirical heuristics have broad implications for scalable information dissemination, knowledge retrieval, and intervention design in large-scale complex networks (Wang et al., 2 Nov 2025, Nguyen et al., 2012, Chen et al., 2022, Cordasco et al., 2015).