Approximation, FPT, and kernelization for directed subset feedback problems

Determine whether any of Directed Feedback Arc Set, Subset-Directed Feedback Arc Set, Directed Feedback Vertex Set, or Subset-Directed Feedback Vertex Set admits a constant-factor approximation algorithm, an FPT algorithm with running time c^k n^{O(1)} for a constant c, or a polynomial kernel.

Background

The paper places Subset-FAST in the broader family of directed feedback arc and feedback vertex problems. For a directed graph, the subset variants require eliminating only cycles that contain at least one designated terminal vertex, whereas the non-subset variants require eliminating all directed cycles. The authors note that, despite known approximation and fixed-parameter tractability results, several fundamental algorithmic questions remain unresolved for all four general directed problems.

References

However, it remains unknown whether any of {\sc DFAS}, {\sc Subset-DFAS}, {\sc DFVS}, or {\sc Subset-DFVS} admit a constant factor approximation, a $ck n{(1)}$ time FPT algorithm, or a polynomial kernel.

— A Quadratic Vertex Kernel and a Subexponential Algorithm for Subset-FAST  (2503.07208 - Jana et al., 10 Mar 2025) in Section 1, Introduction