Subset Feedback Vertex Set is fixed-parameter tractable Marek Cygan, - - PowerPoint PPT Presentation

Subset Feedback Vertex Set is fixed-parameter tractable

Marek Cygan, Marcin Pilipczuk, Micha l Pilipczuk, Jakub Onufry Wojtaszczyk

Institute of Informatics University of Warsaw Google Inc., Cracow, Poland

06 July 2011, Zurich

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 1/18

Feedback Vertex Set

Feedback Vertex Set Input: A graph G = (V , E) and an integer k. Question: Does there exist X ⊆ V , |X| ≤ k, such that G \ X is a forest? In other words, X hits all cycles of G.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 2/18

Feedback Vertex Set

Feedback Vertex Set Input: A graph G = (V , E) and an integer k. Question: Does there exist X ⊆ V , |X| ≤ k, such that G \ X is a forest? In other words, X hits all cycles of G.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 2/18

Feedback Vertex Set

Feedback Vertex Set Input: A graph G = (V , E) and an integer k. Question: Does there exist X ⊆ V , |X| ≤ k, such that G \ X is a forest? In other words, X hits all cycles of G.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 2/18

Feedback Vertex Set

Feedback Vertex Set Input: A graph G = (V , E) and an integer k. Question: Does there exist X ⊆ V , |X| ≤ k, such that G \ X is a forest? In other words, X hits all cycles of G.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 2/18

Feedback Vertex Set

Feedback Vertex Set Input: A graph G = (V , E) and an integer k. Question: Does there exist X ⊆ V , |X| ≤ k, such that G \ X is a forest? In other words, X hits all cycles of G. NP-hard, can be solved in deterministic time 3.83knO(1) [CCL10]. Classical problem with multiple applications. On Karp’s list of 21 NP-hard problems.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 2/18

Subset Feedback Vertex Set

Subset FVS Input: G = (V , E), a set S ⊆ V of red vertices and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red vertex?

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 3/18

Subset Feedback Vertex Set

Subset FVS Input: G = (V , E), a set S ⊆ V of red vertices and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red vertex?

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 3/18

Subset Feedback Vertex Set

Subset FVS Input: G = (V , E), a set S ⊆ V of red vertices and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red vertex?

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 3/18

Subset Feedback Vertex Set

Subset FVS Input: G = (V , E), a set S ⊆ V of red vertices and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red vertex?

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 3/18

Subset Feedback Vertex Set

Subset FVS Input: G = (V , E), a set S ⊆ V of red vertices and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red vertex? A generalization of both FVS and Multiway Cut. Closely related to graph-separation problems. Applications in genetic linkage [ENZ96].

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 3/18

Our result

Subset FVS Input: G = (V , E), a set S ⊆ V of red vertices and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red vertex? Theorem (Cygan, P., Pilipczuk, Wojtaszczyk) Subset Feedback Vertex Set can be solved in time 2O(k log k)nO(1).

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 4/18

SFVS: small change

Subset FVS Input: G = (V , E), a set S ⊆ V of red vertices and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red vertex?

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 5/18

SFVS: small change

Edge Subset FVS Input: G = (V , E), a set S ⊆ E of red edges and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red edge?

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 5/18

SFVS: small change

Edge Subset FVS Input: G = (V , E), a set S ⊆ E of red edges and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red edge? Equivalent!

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 5/18

ESFVS: |S| as a parameter

Edge Subset FVS Input: G = (V , E), a set S ⊆ E of red edges and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red edge? Let us solve a simpler problem.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 6/18

ESFVS: |S| as a parameter

Edge Subset FVS Input: G = (V , E), a set S ⊆ E of red edges and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red edge? Let us solve a simpler problem. Note that if |S| ≤ k, the instance is trivial.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 6/18

ESFVS: |S| as a parameter

Edge Subset FVS Input: G = (V , E), a set S ⊆ E of red edges and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red edge? Let us solve a simpler problem. Note that if |S| ≤ k, the instance is trivial. Does there exist a f (|S|)nO(1) algorithm?

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 6/18

ESFVS: |S| as a parameter

Edge Subset FVS Input: G = (V , E), a set S ⊆ E of red edges and k. Question: Does there exist X ⊆ V that hits all cycles that pass through at least one red edge? Let us solve a simpler problem. Note that if |S| ≤ k, the instance is trivial. Does there exist a f (|S|)nO(1) algorithm? If we do not care about f , quite easy!

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 6/18

SFVS: |S| as a parameter

X solution Assume we have a solution X.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 7/18

SFVS: |S| as a parameter

X solution Assume we have a solution X. In G \ X all red edges are bridges.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 7/18

SFVS: |S| as a parameter

X solution Assume we have a solution X. In G \ X all red edges are bridges. A bubble — a connected component of G \ X \ S.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 7/18

SFVS: |S| as a parameter

X solution Assume we have a solution X. In G \ X all red edges are bridges. A bubble — a connected component of G \ X \ S. Bubbles + red edges = bubble forest.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 7/18

SFVS: |S| as a parameter

X solution Assume we have a solution X. In G \ X all red edges are bridges. A bubble — a connected component of G \ X \ S. Bubbles + red edges = bubble forest. Guess the shape of the bubble forest (rougly |S||S| choices).

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 7/18

SFVS: |S| as a parameter

X solution Assume we have a solution X. In G \ X all red edges are bridges. A bubble — a connected component of G \ X \ S. Bubbles + red edges = bubble forest. Guess the shape of the bubble forest (rougly |S||S| choices). The rest is an instance of Multiway Cut, solvable in 4knO(1) [CLL07].

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 7/18

SFVS: second part

Using techniques of Guillemot [G08], we can do a bit better. Theorem (Cygan, P., Pilipczuk, Wojtaszczyk) SFVS can be solved in time |S|O(k)nO(1).

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 8/18

SFVS: second part

Using techniques of Guillemot [G08], we can do a bit better. Theorem (Cygan, P., Pilipczuk, Wojtaszczyk) SFVS can be solved in time |S|O(k)nO(1). Now our goal is to reduce |S| to O(k3).

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 8/18

SFVS: second part

Using techniques of Guillemot [G08], we can do a bit better. Theorem (Cygan, P., Pilipczuk, Wojtaszczyk) SFVS can be solved in time |S|O(k)nO(1). Now our goal is to reduce |S| to O(k3). This gives the desired time complexity 2O(k log k)nO(1).

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 8/18

SFVS: second part

Using techniques of Guillemot [G08], we can do a bit better. Theorem (Cygan, P., Pilipczuk, Wojtaszczyk) SFVS can be solved in time |S|O(k)nO(1). Now our goal is to reduce |S| to O(k3). This gives the desired time complexity 2O(k log k)nO(1). We use iterative compression, thus we can assume that we are given a solution Z ⊆ V of size k + 1.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 8/18

SFVS: reduce |S|

Z (k + 1)-solution Z is a solution of size k + 1.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 9/18

SFVS: reduce |S|

Z (k + 1)-solution Z is a solution of size k + 1. The graph G \ Z is a bubble forest!

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 9/18

SFVS: reduce |S|

Z (k + 1)-solution Z is a solution of size k + 1. The graph G \ Z is a bubble forest! Two types of red edges: incident with Z and bridges in the bubble forest.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 9/18

SFVS: red edges of the first type

Z (k + 1)-solution Bound on the first type: red edges incident with Z.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 10/18

SFVS: red edges of the first type

Z (k + 1)-solution Bound on the first type: red edges incident with Z. We provide a reduction that reduces a red degree of a vertex to 10k.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 10/18

SFVS: red edges of the first type

Z (k + 1)-solution Bound on the first type: red edges incident with Z. We provide a reduction that reduces a red degree of a vertex to 10k. The reduction goes along the lines of the quadratic kernel for Feedback Vertex Set due to Thomasse (2009).

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 10/18

SFVS: red edges of the first type

Z (k + 1)-solution Bound on the first type: red edges incident with Z. We provide a reduction that reduces a red degree of a vertex to 10k. The reduction goes along the lines of the quadratic kernel for Feedback Vertex Set due to Thomasse (2009). |Z| = k + 1 ⇒ O(k2) red edges of the first type.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 10/18

SFVS: red edges of the second type

Bound on the second type: bridges in the bubble forest.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 11/18

SFVS: red edges of the second type

Bound on the second type: bridges in the bubble forest. There are three types of bubbles:

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 11/18

SFVS: red edges of the second type

Bound on the second type: bridges in the bubble forest. There are three types of bubbles: leaf bubble zero or one neighbouring bubbles,

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 11/18

SFVS: red edges of the second type

Bound on the second type: bridges in the bubble forest. There are three types of bubbles: leaf bubble zero or one neighbouring bubbles, edge bubble two neighbouring bubbles,

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 11/18

SFVS: red edges of the second type

Bound on the second type: bridges in the bubble forest. There are three types of bubbles: leaf bubble zero or one neighbouring bubbles, edge bubble two neighbouring bubbles, internal bubble at least tree neighbouring bubbles.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 11/18

SFVS: bridge red edges

Number of internal bubbles ≤ number of leaf bubbles.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 12/18

SFVS: bridge red edges

Number of internal bubbles ≤ number of leaf bubbles. Number of paths of edge bubbles ≤ 2× number of leaf bubbles.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 12/18

SFVS: bridge red edges

Number of internal bubbles ≤ number of leaf bubbles. Number of paths of edge bubbles ≤ 2× number of leaf bubbles. Need two bounds: number of leaf bubbles and total length of all paths of edge bubbles.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 12/18

SFVS: edge bubbles bound

|Z| = k + 1 Sketch of the bound of total length of paths of edge bubbles.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 13/18

SFVS: edge bubbles bound

|Z| = k + 1 Sketch of the bound of total length of paths of edge bubbles. Edge bubbles look like on the figure.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 13/18

SFVS: edge bubbles bound

|Z| = k + 1 Sketch of the bound of total length of paths of edge bubbles. Edge bubbles look like on the figure.

Degree-2 edge bubbles can be replaced by a single red edge.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 13/18

SFVS: edge bubbles bound

|Z| = k + 1 Sketch of the bound of total length of paths of edge bubbles. Edge bubbles look like on the figure.

Degree-2 edge bubbles can be replaced by a single red edge.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 13/18

SFVS: edge bubbles bound

|Z| = k + 1 Sketch of the bound of total length of paths of edge bubbles. Edge bubbles look like on the figure.

Degree-2 edge bubbles can be replaced by a single red edge. From definition: extra edges incident to edge bubble must go to Z.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 13/18

SFVS: edge bubbles bound

|Z| = k + 1 We have long paths of edge bubbles, each neighbouing Z.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 14/18

SFVS: edge bubbles bound

|Z| = k + 1 We have long paths of edge bubbles, each neighbouing Z. Long paths of edge bubbles ⇒ a lot of pairs of neighbouring edge bubbles.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 14/18

SFVS: edge bubbles bound

|Z| = k + 1 We have long paths of edge bubbles, each neighbouing Z. Long paths of edge bubbles ⇒ a lot of pairs of neighbouring edge bubbles. Only k such pairs can be hit by the solution.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 14/18

SFVS: edge bubbles bound

|Z| = k + 1 |Z| such not hit pairs together Z makes a cycle with a red edge.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 15/18

SFVS: edge bubbles bound

|Z| = k + 1 |Z| such not hit pairs together Z makes a cycle with a red edge.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 15/18

SFVS: edge bubbles bound

|Z| = k + 1 |Z| such not hit pairs together Z makes a cycle with a red edge. Too long edge bubbles paths ⇒ too much not hit pairs of edge bubbles ⇒ a solution needs to take a vertex from Z.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 15/18

SFVS: edge bubbles bound

|Z| = k + 1 |Z| such not hit pairs together Z makes a cycle with a red edge. Too long edge bubbles paths ⇒ too much not hit pairs of edge bubbles ⇒ a solution needs to take a vertex from Z. We guess solution vertices from Z ⇒ 2k+1 subinstances in total.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 15/18

SFVS: summary

To sum up: At each step of iterative compression,

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 16/18

SFVS: summary

To sum up: At each step of iterative compression, we reduce to at most 2k+1 instances with |S| = O(k3),

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 16/18

SFVS: summary

To sum up: At each step of iterative compression, we reduce to at most 2k+1 instances with |S| = O(k3),

reduce the red degree to 10k ⇒ O(k2) red edges incident with Z;

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 16/18

SFVS: summary

To sum up: At each step of iterative compression, we reduce to at most 2k+1 instances with |S| = O(k3),

reduce the red degree to 10k ⇒ O(k2) red edges incident with Z; reduce the number of leaf bubbles, and

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 16/18

SFVS: summary

To sum up: At each step of iterative compression, we reduce to at most 2k+1 instances with |S| = O(k3),

reduce the red degree to 10k ⇒ O(k2) red edges incident with Z; reduce the number of leaf bubbles, and reduce the number of edge bubbles

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 16/18

SFVS: summary

To sum up: At each step of iterative compression, we reduce to at most 2k+1 instances with |S| = O(k3),

reduce the red degree to 10k ⇒ O(k2) red edges incident with Z; reduce the number of leaf bubbles, and reduce the number of edge bubbles ⇒ O(k3) red edges that are bridges in the bubble forest.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 16/18

SFVS: summary

To sum up: At each step of iterative compression, we reduce to at most 2k+1 instances with |S| = O(k3),

reduce the red degree to 10k ⇒ O(k2) red edges incident with Z; reduce the number of leaf bubbles, and reduce the number of edge bubbles ⇒ O(k3) red edges that are bridges in the bubble forest.

Each instance we solve with |S|O(k)nO(1) time algorithm.

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 16/18

SFVS: open questions

Open questions: Polynomial kernel for SFVS?

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 17/18

SFVS: open questions

Open questions: Polynomial kernel for SFVS? Improve 8-approximation [ENZ96]. Should be 2. . .

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 17/18

Thank you

Questions?

Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 18/18