Liu and Tao prove influence maximization APX-hard on DAGs
No polynomial-time approximation exists for Linear Threshold model on DAGs...
A new theoretical paper by Panfeng Liu and Biaoshuai Tao tackles the computational limits of influence maximization, a key problem in viral marketing and network analysis. The authors show that under the Linear Threshold (LT) model, finding an optimal set of seed nodes remains APX-hard even on Directed Acyclic Graphs (DAGs) — a surprising result because DAGs lack cycles that often cause complexity. This means no polynomial-time approximation scheme (PTAS) exists for DAGs under LT, forcing researchers to look for even more restricted topologies.
On the positive side, for arborescences (directed trees with a single root or sink), the problem becomes tractable. For out-arborescences, the IC and LT models are shown to be equivalent, and exact dynamic programming algorithms run in polynomial time. For in-arborescences, LT is already polynomial-time solvable, and the authors contribute a fully polynomial-time approximation scheme (FPTAS) for the Independent Cascade (IC) model. These results provide both hardness boundaries and practical solution pathways for real-world influence maximization on tree-like networks.
- Influence maximization under the Linear Threshold model is APX-hard on Directed Acyclic Graphs, meaning no PTAS exists even without cycles.
- For out-arborescences, IC and LT models become equivalent, enabling exact polynomial-time dynamic programming algorithms.
- A new FPTAS is provided for the IC model on in-arborescences, offering near-optimal approximations in polynomial time.
Why It Matters
This work clarifies which network structures allow efficient viral marketing, guiding algorithm choice in social and information networks.