# Feedback vertex set

In the mathematical discipline of graph theory, a feedback vertex set of a graph is a set of vertices whose removal leaves a graph without cycles. In other words, each feedback vertex set contains at least one vertex of any cycle in the graph. The feedback vertex set problem is an NP-complete problem in computational complexity theory. It was among the first problems shown to be NP-complete. It has wide applications in operating systems, database systems, and VLSI chip design.

## Definition

The decision problem is as follows:

INSTANCE: An (undirected or directed) graph $G = (V, E)$ and a positive integer $k$.
QUESTION: Is there a subset $X \subseteq V$ with $|X| \leq k$ such that $G$ with the vertices from $X$ deleted is cycle-free?

The graph $G[V \setminus X]$ that remains after removing $X$ from $G$ is an induced forest (resp. an induced directed acyclic graph in the case of directed graphs). Thus, finding a minimum feedback vertex set in a graph is equivalent to finding a maximum induced forest (resp. maximum induced directed acyclic graph in the case of directed graphs).

## NP-hardness

Karp (1972) showed that the feedback vertex set problem for directed graphs is NP-complete. The problem remains NP-complete on directed graphs with maximum in-degree and out-degree two, and on directed planar graphs with maximum in-degree and out-degree three.[1] Karp's reduction also implies the NP-completeness of the feedback vertex set problem on undirected graphs, where the problem stays NP-hard on graphs of maximum degree four. The feedback vertex set problem can be solved in polynomial time on graphs of maximum degree at most three.[2]

Note that the problem of deleting as few edges as possible to make the graph cycle-free is equivalent to finding a spanning tree, which can be done in polynomial time. In contrast, the problem of deleting edges from a directed graph to make it acyclic, the feedback arc set problem, is NP-complete.[3]

## Exact algorithms

The corresponding NP optimization problem of finding the size of a minimum feedback vertex set can be solved in time O(1.7347n), where n is the number of vertices in the graph.[4] This algorithm actually computes a maximum induced forest, and when such a forest is obtained, its complement is a minimum feedback vertex set. The number of minimal feedback vertex sets in a graph is bounded by O(1.8638n).[5] The directed feedback vertex set problem can still be solved in time O*(1.9977n), where n is the number of vertices in the given directed graph.[6] The parameterized versions of the directed and undirected problems are both fixed-parameter tractable.[7]

## Approximation

The problem is APX-complete, which directly follows from the APX-completeness of the vertex cover problem,[8] and the existence of an approximation preserving L-reduction from the vertex cover problem to it.[3] The best known approximation algorithm on undirected graphs is by a factor of two.[9]

## Bounds

According to the Erdős–Pósa theorem, the size of a minimum feedback vertex set is within a logarithmic factor of the maximum number of vertex-disjoint cycles in the given graph.[10]

## Applications

In operating systems, feedback vertex sets play a prominent role in the study of deadlock recovery. In the wait-for graph of an operating system, each directed cycle corresponds to a deadlock situation. In order to resolve all deadlocks, some blocked processes have to be aborted. A minimum feedback vertex set in this graph corresponds to a minimum number of processes that one needs to abort.[11]

Furthermore, the feedback vertex set problem has applications in VLSI chip design.[12]

## Notes

1. ^ unpublished results due to Garey and Johnson, cf. Garey & Johnson (1979): GT7
2. ^
3. ^ a b
4. ^
5. ^
6. ^
7. ^
8. ^ Dinur & Safra 2005
9. ^ Becker & Geiger (1996). See also Bafna, Berman & Fujito (1999) for an alternative approximation algorithm with the same approximation ratio.
10. ^
11. ^
12. ^