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 . 83 k n O (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 2 O ( k log k ) n O (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 | ) n O (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 | ) n O (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 solution X Assume we have a solution X . Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 7/18

SFVS: | S | as a parameter solution X 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 solution X 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 solution X 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 solution X 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 solution X 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 4 k n O (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 ) n O (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 ) n O (1) . Now our goal is to reduce | S | to O ( k 3 ). 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 ) n O (1) . Now our goal is to reduce | S | to O ( k 3 ). This gives the desired time complexity 2 O ( k log k ) n O (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 ) n O (1) . Now our goal is to reduce | S | to O ( k 3 ). This gives the desired time complexity 2 O ( k log k ) n O (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 | ( k + 1)-solution Z Z is a solution of size k + 1. Cygan, Pilipczuk, Pilipczuk, Wojtaszczyk Subset Feedback Vertex Set is FPT 9/18