Hardness of Vertex Deletion and Project Scheduling

Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer k ≥ 2 and arbitrary small e > 0, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor k − e even on graphs where the vertices can be almost partitioned into k solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the “It Ain’t Over Till It’s Over” theorem) than the previous hardness result.