On Deleting Vertices to Reduce Density in Graphs and Supermodular Functions

We consider deletion problems in graphs and supermodular functions where the goal is to reduce density. In Graph Density Deletion (GraphDD), we are given a graph $G=(V,E)$ with non-negative vertex costs and a non-negative parameter $\rho \ge 0$ and the goal is to remove a minimum cost subset $S$ of vertices such that the densest subgraph in $G-S$ has density at most $\rho$. This problem has an underlying matroidal structure and generalizes several classical problems such as vertex cover, feedback vertex set, and pseudoforest deletion set for appropriately chosen $\rho \le 1$ and all of these classical problems admit a $2$-approximation. In sharp contrast, we prove that for every fixed integer $\rho > 1$, GraphDD is hard to approximate to within a logarithmic factor via a reduction from Set Cover, thus showing a phase transition phenomenon. Next, we investigate a generalization of GraphDD to monotone supermodular functions, termed Supermodular Density Deletion (SupmodDD). In SupmodDD, we are given a monotone supermodular function $f:2^V \rightarrow \mathbb{Z}_{\ge 0}$ via an evaluation oracle with element costs and a non-negative integer $\rho \ge 0$ and the goal is remove a minimum cost subset $S \subseteq V$ such that the densest subset according to $f$ in $V-S$ has density at most $\rho$. We show that SupmodDD is approximation equivalent to the well-known Submodular Cover problem; this implies a tight logarithmic approximation and hardness for SupmodDD; it also implies a logarithmic approximation for GraphDD, thus matching our inapproximability bound. Motivated by these hardness results, we design bicriteria approximation algorithms for both GraphDD and SupmodDD.