Single-Exponential Algorithms for Directed Feedback Vertex Set on Planar Digraphs

2026-09-26

AI Summary

This study addresses the long-standing double-exponential time bottleneck in parameterized algorithms for the Feedback Vertex Set (FVS) problem on planar directed graphs, which has historically been constrained by general-graph complexities. This work proposes the first single-exponential fixed-parameter tractable (FPT) algorithm for this problem. The core methodology leverages Euler’s counting identity and planar embedding theory to reduce FVS to the polynomially solvable Feedback Arc Set problem, subsequently designing both randomized and deterministic solving strategies based on the Lucchesi–Younger theorem. The primary contribution is a breakthrough achievement of single-exponential time complexity, with the randomized and deterministic algorithms running in O*(4.24^k) and O*(8.04^k) time, respectively. Both approaches require only polynomial space, substantially improving computational efficiency for this problem.

Abstract

We consider Directed Feedback Vertex Set on planar digraphs, parameterized by the solution size $k$. We give a randomized algorithm with one-sided error running in time $(2+\sqrt5)^k n^{O(1)}= 4.24^k n^{O(1)}$, and a deterministic algorithm running in time $8.04^k n^{O(1)}$. Both algorithms use polynomial space. To the best of our knowledge, these are the first single-exponential fixed-parameter algorithms for Directed Feedback Vertex Set on planar digraphs. This contrasts with general digraphs, where the best known algorithms run in time $2^{O(k\log k)}(n+m)$, and whether a $2^{o(k\log k)}n^{O(1)}$-time algorithm exists remains a major open problem. Our main tool is an exact Euler-type counting identity for plane digraphs. It shows that every small solution must carry a large share of the vertices whose in- and out-arcs alternate in the embedding, while solutions avoiding such vertices can be computed by reducing to Directed Feedback Arc Set, which is known to be solvable in polynomial time on planar digraphs via the Lucchesi-Younger theorem.

Problem

• Directed Feedback Vertex Set
• Planar Digraphs
• Fixed-Parameter Tractability
• Single-Exponential Algorithm

Innovation

• Directed Feedback Vertex Set
• Planar Digraphs
• Single-Exponential Algorithm
• Fixed-Parameter Tractability
• Euler-type Counting Identity