Extracting Densest Sub-hypergraph with Convex Edge-Weight Functions

The densest subgraph problem (DSG) aiming at finding an induced such that the average edge-weights of is maximized, a well-studied problem. However, when input graph hypergraph, existing notion DSG fails to capture fact hyperedge partially belonging sub-hypergraph also part sub-hypergraph. To resolve issue, we suggest function $$f_e:\mathbb {Z}_{\ge 0}\rightarrow \mathbb {R}_{\ge 0}$$ represent partial edge-weight e in hypergraph $$\mathcal {H}=(V,\mathcal {E},f)$$ and formulate generalized (GDSH) as $$\max _{S\subseteq V}\frac{\sum _{e\in \mathcal {E}}{f_e(|e\cap S|)}}{|S|}$$ . We demonstrate that, all functions are non-decreasing convex, GDSH can be solved polynomial-time by linear program-based algorithm, network flow-based algorithm greedy $$\frac{1}{r}$$ -approximation where r rank hypergraph. Finally, investigate computational tractability some non-convex.