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Polynomial-Time Algorithms for the Subset Feedback Vertex Set - - PowerPoint PPT Presentation

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Polynomial-Time Algorithms for the Subset Feedback Vertex Set Problem on Interval Graphs and Permutation Graphs Charis Papadopoulos Spyridon Tzimas 21st International Symposium on Fundamentals of Computation Theory - FCT 2017 Bordeaux, France,

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Polynomial-Time Algorithms for the Subset Feedback Vertex Set Problem on Interval Graphs and Permutation Graphs

Charis Papadopoulos Spyridon Tzimas

21st International Symposium on Fundamentals of Computation Theory - FCT 2017 Bordeaux, France, September 2017

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Feedback Vertex Set (FVS)

Feedback Vertex Set – FVS Input: A graph G = (V , E) Output: Find a set X ⊂ V of minimum cardinality such that G − X is acyclic (forest). G G − X

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 2 / 21

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Feedback Vertex Set (FVS)

Feedback Vertex Set – FVS Input: A graph G = (V , E) Output: Find a set X ⊂ V of minimum cardinality such that G − X is acyclic (forest). G G − X G − X ′

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 2 / 21

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Feedback Vertex Set (FVS)

Feedback Vertex Set – FVS Input: A graph G = (V , E) Output: Find a set X ⊂ V of minimum cardinality such that G − X is acyclic (forest). G G − X G − X ′ Weighted FVS: Weights on V → minimize

  • v∈X

w(v)

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 2 / 21

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Subset Feedback Vertex Set (SFVS)

Subset Feedback Vertex Set – SFVS Input: A graph G = (V , E) and a vertex set S ⊆ V Output: Find a set X ⊂ V of minimum cardinality such that no cycle of G − X contains a vertex of S. G G − X

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 3 / 21

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Subset Feedback Vertex Set (SFVS)

Subset Feedback Vertex Set – SFVS Input: A graph G = (V , E) and a vertex set S ⊆ V Output: Find a set X ⊂ V of minimum cardinality such that no cycle of G − X contains a vertex of S. G G − X G − X ′

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 3 / 21

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Subset Feedback Vertex Set (SFVS)

Subset Feedback Vertex Set – SFVS Input: A graph G = (V , E) and a vertex set S ⊆ V Output: Find a set X ⊂ V of minimum cardinality such that no cycle of G − X contains a vertex of S. G G − X G − X ′ Weighted SFVS: Weights on V → minimize

  • v∈X

w(v).

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 3 / 21

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SLIDE 8

Subset Feedback Vertex Set (SFVS)

Subset Feedback Vertex Set – SFVS Input: A graph G = (V , E) and a vertex set S ⊆ V Output: Find a set X ⊂ V of minimum cardinality such that no cycle of G − X contains a vertex of S. G G − X G − X ′ If S = V = ⇒ SFVS ≡ FVS.

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 3 / 21

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Subset Feedback Vertex Set (SFVS)

Subset Feedback Vertex Set – SFVS Input: A graph G = (V , E) and a vertex set S ⊆ V Output: Find a set X ⊂ V of minimum cardinality such that no cycle of G − X contains a vertex of S. G G − X G − X ′ If S = V = ⇒ SFVS ≡ FVS. If S = {∅} = ⇒ X = ∅.

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 3 / 21

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Previous Results on FVS and SFVS

Both problems are NP-complete (Garey and Johnson, ’79)

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 4 / 21

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Previous Results on FVS and SFVS

Both problems are NP-complete (Garey and Johnson, ’79) Exact algorithms

FVS: O(1.75n) (Raman et al., ’08), O(1.86n) (weighted, Fomin et al., ’08) SFVS: O(1.76n) (Fomin et al., ’16), O(1.86n) (weighted, Fomin et al., ’13) SFVS: chordal O(1.68n) (Golovach, ’14), AT-free O(1.62n) (Chitnis, ’17)

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 4 / 21

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Previous Results on FVS and SFVS

Both problems are NP-complete (Garey and Johnson, ’79) Exact algorithms

FVS: O(1.75n) (Raman et al., ’08), O(1.86n) (weighted, Fomin et al., ’08) SFVS: O(1.76n) (Fomin et al., ’16), O(1.86n) (weighted, Fomin et al., ’13) SFVS: chordal O(1.68n) (Golovach, ’14), AT-free O(1.62n) (Chitnis, ’17)

Restricted to graph classes

FVS NP-complete: bipartite and planar FVS ∈ P: chordal (Spinrad, 2003), AT-free (Kratsch et al., 2008)

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 4 / 21

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Previous Results on FVS and SFVS

Both problems are NP-complete (Garey and Johnson, ’79) Exact algorithms

FVS: O(1.75n) (Raman et al., ’08), O(1.86n) (weighted, Fomin et al., ’08) SFVS: O(1.76n) (Fomin et al., ’16), O(1.86n) (weighted, Fomin et al., ’13) SFVS: chordal O(1.68n) (Golovach, ’14), AT-free O(1.62n) (Chitnis, ’17)

Restricted to graph classes

FVS NP-complete: bipartite and planar FVS ∈ P: chordal (Spinrad, 2003), AT-free (Kratsch et al., 2008) SFVS NP-complete: split (Fomin et al., 2013)

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 4 / 21

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Previous Results on FVS and SFVS

Both problems are NP-complete (Garey and Johnson, ’79) Exact algorithms

FVS: O(1.75n) (Raman et al., ’08), O(1.86n) (weighted, Fomin et al., ’08) SFVS: O(1.76n) (Fomin et al., ’16), O(1.86n) (weighted, Fomin et al., ’13) SFVS: chordal O(1.68n) (Golovach, ’14), AT-free O(1.62n) (Chitnis, ’17)

Restricted to graph classes

FVS NP-complete: bipartite and planar FVS ∈ P: chordal (Spinrad, 2003), AT-free (Kratsch et al., 2008) SFVS NP-complete: split (Fomin et al., 2013) SFVS ∈ P: ?

AT-free

SFVS:?

chordal

SFVS:NP-c

co-bipartite

SFVS:?

permutation

SFVS:?

interval

SFVS:?

split

SFVS:NP-c FVS:P

⊃ ⊃ ⊂ ⊂ ⊃

1

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 4 / 21

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Our Results

Both problems are NP-complete (Garey and Johnson, ’79) Exact algorithms

FVS: O(1.75n) (Raman et al., ’08), O(1.86n) (weighted, Fomin et al., ’08) SFVS: O(1.76n) (Fomin et al., ’16), O(1.86n) (weighted, Fomin et al., ’13) SFVS: chordal O(1.68n) (Golovach, ’14), AT-free O(1.62n) (Chitnis, ’17)

Restricted to graph classes

FVS NP-complete: bipartite and planar FVS ∈ P: chordal (Spinrad, 2003), AT-free (Kratsch et al., 2008) SFVS NP-complete: split (Fomin et al., 2013) SFVS ∈ P: co-bipartite, interval, permutation

AT-free

SFVS:?

chordal

SFVS:NP-c

co-bipartite

SFVS: P

permutation

SFVS: P

interval

SFVS: P

split

SFVS:NP-c FVS:P

⊃ ⊃ ⊂ ⊂ ⊃

1

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 5 / 21

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Maximal S-forests

An SFVS X is minimal if no set X ′ ⊂ X is an SFVS. G G − X

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 6 / 21

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Maximal S-forests

An SFVS X is minimal if no set X ′ ⊂ X is an SFVS. G G − X Every vertex v of a minimal X: is the unique that is deleted from some S-cycle ⇒ Cv (certifying cycle)

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 6 / 21

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Maximal S-forests

An SFVS X is minimal if no set X ′ ⊂ X is an SFVS. G G − X Every vertex v of a minimal X: is the unique that is deleted from some S-cycle ⇒ Cv (certifying cycle) If X is a minimal SFVS: Y = V − X is a maximal S-forest FS: the set of maximal S-forests

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 6 / 21

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Maximal S-forests

An SFVS X is minimal if no set X ′ ⊂ X is an SFVS. G G − X Every vertex v of a minimal X: is the unique that is deleted from some S-cycle ⇒ Cv (certifying cycle) If X is a minimal SFVS: Y = V − X is a maximal S-forest FS: the set of maximal S-forests The maximum solution is among FS ⇒ Enumerate all maximal S-forests

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 6 / 21

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Complements of bipartite graphs

co-bipartite graphs: V is partitioned into two cliques A and B A B

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 7 / 21

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Complements of bipartite graphs

co-bipartite graphs: V is partitioned into two cliques A and B A B SA SB SA=A ∩ S and SB=B ∩ S

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 7 / 21

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Complements of bipartite graphs

co-bipartite graphs: V is partitioned into two cliques A and B A B SA SB SA=A ∩ S and SB=B ∩ S For any maximal S-forest F ⇒ |F ∩ SA| ≤ 2 and |F ∩ SB| ≤ 2

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 7 / 21

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Complements of bipartite graphs

co-bipartite graphs: V is partitioned into two cliques A and B A B SA SB SA=A ∩ S and SB=B ∩ S For any maximal S-forest F ⇒ |F ∩ SA| ≤ 2 and |F ∩ SB| ≤ 2 There are at most 22n4 maximal S-forests

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 7 / 21

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SLIDE 24

Complements of bipartite graphs

co-bipartite graphs: V is partitioned into two cliques A and B A B SA SB SA=A ∩ S and SB=B ∩ S For any maximal S-forest F ⇒ |F ∩ SA| ≤ 2 and |F ∩ SB| ≤ 2 There are at most 22n4 maximal S-forests ⇒ An O(n4) algorithm for computing SFVS on co-biparite graphs.

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 7 / 21

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Interval Graphs

a b c d e f g I: representation of closed intervals interval graphs: V ↔ I such that (u, v) ∈ E iff u and v intersect Every induced cycle of an interval graph is a triangle

a b c d e f g 1

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 8 / 21

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Interval Graphs

a b c d e f g I: representation of closed intervals interval graphs: V ↔ I such that (u, v) ∈ E iff u and v intersect Every induced cycle of an interval graph is a triangle Compute maximal S-forests FS based on I Dynamic programming approach

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 8 / 21

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SLIDE 27

Interval Graphs

a b c d e f g I: representation of closed intervals interval graphs: V ↔ I such that (u, v) ∈ E iff u and v intersect Every induced cycle of an interval graph is a triangle Compute maximal S-forests FS based on I Dynamic programming approach Scan vertices from left-to-right according to their right endpoint

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 8 / 21

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Interval Graphs

Va

a b c d e f g I: representation of closed intervals interval graphs: V ↔ I such that (u, v) ∈ E iff u and v intersect Every induced cycle of an interval graph is a triangle Compute maximal S-forests FS based on I Dynamic programming approach Scan vertices from left-to-right according to their right endpoint We grow appropriately each maximal S-forest

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 8 / 21

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Interval Graphs

Vb

a b c d e f g I: representation of closed intervals interval graphs: V ↔ I such that (u, v) ∈ E iff u and v intersect Every induced cycle of an interval graph is a triangle Compute maximal S-forests FS based on I Dynamic programming approach Scan vertices from left-to-right according to their right endpoint We grow appropriately each maximal S-forest

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 8 / 21

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Interval Graphs

Vc

a b c d e f g I: representation of closed intervals interval graphs: V ↔ I such that (u, v) ∈ E iff u and v intersect Every induced cycle of an interval graph is a triangle Compute maximal S-forests FS based on I Dynamic programming approach Scan vertices from left-to-right according to their right endpoint We grow appropriately each maximal S-forest

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 8 / 21

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SLIDE 31

Interval Graphs

Vd

a b c d e f g I: representation of closed intervals interval graphs: V ↔ I such that (u, v) ∈ E iff u and v intersect Every induced cycle of an interval graph is a triangle Compute maximal S-forests FS based on I Dynamic programming approach Scan vertices from left-to-right according to their right endpoint We grow appropriately each maximal S-forest

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 8 / 21

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SLIDE 32

Interval Graphs

Ve

a b c d e f g I: representation of closed intervals interval graphs: V ↔ I such that (u, v) ∈ E iff u and v intersect Every induced cycle of an interval graph is a triangle Compute maximal S-forests FS based on I Dynamic programming approach Scan vertices from left-to-right according to their right endpoint We grow appropriately each maximal S-forest

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 8 / 21

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SLIDE 33

Interval Graphs

Vf

a b c d e f g I: representation of closed intervals interval graphs: V ↔ I such that (u, v) ∈ E iff u and v intersect Every induced cycle of an interval graph is a triangle Compute maximal S-forests FS based on I Dynamic programming approach Scan vertices from left-to-right according to their right endpoint We grow appropriately each maximal S-forest

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 8 / 21

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SLIDE 34

Interval Graphs

Vg = V

a b c d e f g I: representation of closed intervals interval graphs: V ↔ I such that (u, v) ∈ E iff u and v intersect Every induced cycle of an interval graph is a triangle Compute maximal S-forests FS based on I Dynamic programming approach Scan vertices from left-to-right according to their right endpoint We grow appropriately each maximal S-forest

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 8 / 21

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SLIDE 35

Predecessors

a b c d e f g The left and right endpoints of the intervals define ≤ℓ and ≤r:

i ℓ(i) r(i) j ℓ(j) r(j) i ≤l j ⇐ ⇒ ℓ(i) ≤ ℓ(j) i ≤r j ⇐ ⇒ r(i) ≤ r(j) 1

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 9 / 21

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SLIDE 36

Predecessors

a b c d e f g The left and right endpoints of the intervals define ≤ℓ and ≤r:

i ℓ(i) r(i) j ℓ(j) r(j) i ≤l j ⇐ ⇒ ℓ(i) ≤ ℓ(j) i ≤r j ⇐ ⇒ r(i) ≤ r(j) 1

Predecessors of an interval i: <i = r- max(Vi \ {i}) the last interval that finishes before i finishes ≪i = r- max(Vi \ ({i} ∪ {h ∈ V : {h, i} ∈ E})) the last interval that finishes before i starts

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 9 / 21

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SLIDE 37

Predecessors

a b c d e f g The left and right endpoints of the intervals define ≤ℓ and ≤r:

i ℓ(i) r(i) j ℓ(j) r(j) i ≤l j ⇐ ⇒ ℓ(i) ≤ ℓ(j) i ≤r j ⇐ ⇒ r(i) ≤ r(j) 1

Predecessors of an interval i: <i = r- max(Vi \ {i}) the last interval that finishes before i finishes ≪i = r- max(Vi \ ({i} ∪ {h ∈ V : {h, i} ∈ E})) the last interval that finishes before i starts

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 9 / 21

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SLIDE 38

Predecessors

Vf

a b c d e f g The left and right endpoints of the intervals define ≤ℓ and ≤r:

i ℓ(i) r(i) j ℓ(j) r(j) i ≤l j ⇐ ⇒ ℓ(i) ≤ ℓ(j) i ≤r j ⇐ ⇒ r(i) ≤ r(j) 1

Predecessors of an interval i: <i = r- max(Vi \ {i}) the last interval that finishes before i finishes ≪i = r- max(Vi \ ({i} ∪ {h ∈ V : {h, i} ∈ E})) the last interval that finishes before i starts

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 9 / 21

slide-39
SLIDE 39

Predecessors

Vf

a b c d <f e f g The left and right endpoints of the intervals define ≤ℓ and ≤r:

i ℓ(i) r(i) j ℓ(j) r(j) i ≤l j ⇐ ⇒ ℓ(i) ≤ ℓ(j) i ≤r j ⇐ ⇒ r(i) ≤ r(j) 1

Predecessors of an interval i: <i = r- max(Vi \ {i}) the last interval that finishes before i finishes ≪i = r- max(Vi \ ({i} ∪ {h ∈ V : {h, i} ∈ E})) the last interval that finishes before i starts

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 9 / 21

slide-40
SLIDE 40

Predecessors

Vf

a b ≪f c d <f e f g The left and right endpoints of the intervals define ≤ℓ and ≤r:

i ℓ(i) r(i) j ℓ(j) r(j) i ≤l j ⇐ ⇒ ℓ(i) ≤ ℓ(j) i ≤r j ⇐ ⇒ r(i) ≤ r(j) 1

Predecessors of an interval i: <i = r- max(Vi \ {i}) the last interval that finishes before i finishes ≪i = r- max(Vi \ ({i} ∪ {h ∈ V : {h, i} ∈ E})) the last interval that finishes before i starts

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 9 / 21

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SLIDE 41

Basic sets: A, B, C

Sets used by our dynamic programming algorithm A: corresponds to a maximum S-forest B and C: choose a solution only from a predescribed set x, y ∈ V − Vi such that x <ℓ y Ai = maxw{X ⊆ Vi : G[X] ∈ FS}

i

Vi

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 10 / 21

slide-42
SLIDE 42

Basic sets: A, B, C

Sets used by our dynamic programming algorithm A: corresponds to a maximum S-forest B and C: choose a solution only from a predescribed set x, y ∈ V − Vi such that x <ℓ y Ai = maxw{X ⊆ Vi : G[X] ∈ FS}

i

Vi

Bx

i = maxw{X ⊆ Vi : G[X ∪ {x}] ∈ FS}

x i

Vi

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 10 / 21

slide-43
SLIDE 43

Basic sets: A, B, C

Sets used by our dynamic programming algorithm A: corresponds to a maximum S-forest B and C: choose a solution only from a predescribed set x, y ∈ V − Vi such that x <ℓ y Ai = maxw{X ⊆ Vi : G[X] ∈ FS}

i

Vi

Bx

i = maxw{X ⊆ Vi : G[X ∪ {x}] ∈ FS}

x i

Vi

C x,y

i

= maxw{X ⊆ Vi : G[X ∪ {x, y}] ∈ FS}

y x i

Vi

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 10 / 21

slide-44
SLIDE 44

Recursive formulation for Ai

A-set: Ai = max

w

  • A<i, Bi

<i ∪ {i}

  • i

Vi

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 11 / 21

slide-45
SLIDE 45

Recursive formulation for Ai

A-set: Ai = max

w

  • A<i, Bi

<i ∪ {i}

  • i

<i

V<i

If i / ∈ Ai then i is irrelevant ⇒ Ai = A<i

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 11 / 21

slide-46
SLIDE 46

Recursive formulation for Ai

A-set: Ai = max

w

  • A<i, Bi

<i ∪ {i}

  • i

<i

V<i

If i / ∈ Ai then i is irrelevant ⇒ Ai = A<i If i ∈ Ai then Bi

<i ∪ {i} contains no S-cycle

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 11 / 21

slide-47
SLIDE 47

Recursive formulation for Bx

i

Let x′ = ℓ- min{i, x} and let y′ = {i, x} \ x′: If {i, x} / ∈ E, then Bx

i = Ai

If {i, x} ∈ E, then Bx

i =

   maxw

  • Bx

<i, Bx′ ≪y′ ∪ {i}

  • , if i or x ∈ S

maxw

  • Bx

<i, C x′,y′ <i

∪ {i}

  • , if i, x /

∈ S x i

Vi

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 12 / 21

slide-48
SLIDE 48

Recursive formulation for Bx

i

Let x′ = ℓ- min{i, x} and let y′ = {i, x} \ x′: If {i, x} / ∈ E, then Bx

i = Ai

If {i, x} ∈ E, then Bx

i =

   maxw

  • Bx

<i, Bx′ ≪y′ ∪ {i}

  • , if i or x ∈ S

maxw

  • Bx

<i, C x′,y′ <i

∪ {i}

  • , if i, x /

∈ S x = x′ i = y′

Vi

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 12 / 21

slide-49
SLIDE 49

Recursive formulation for Bx

i

Let x′ = ℓ- min{i, x} and let y′ = {i, x} \ x′: If {i, x} / ∈ E, then Bx

i = Ai

If {i, x} ∈ E, then Bx

i =

   maxw

  • Bx

<i, Bx′ ≪y′ ∪ {i}

  • , if i or x ∈ S

maxw

  • Bx

<i, C x′,y′ <i

∪ {i}

  • , if i, x /

∈ S x = y′ i = x′

Vi

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 12 / 21

slide-50
SLIDE 50

Recursive formulation for Bx

i

Let x′ = ℓ- min{i, x} and let y′ = {i, x} \ x′: If {i, x} / ∈ E, then Bx

i = Ai

If {i, x} ∈ E, then Bx

i =

   maxw

  • Bx

<i, Bx′ ≪y′ ∪ {i}

  • , if i or x ∈ S

maxw

  • Bx

<i, C x′,y′ <i

∪ {i}

  • , if i, x /

∈ S x i

Vi

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 12 / 21

slide-51
SLIDE 51

Recursive formulation for Bx

i

Let x′ = ℓ- min{i, x} and let y′ = {i, x} \ x′: If {i, x} / ∈ E, then Bx

i = Ai

If {i, x} ∈ E, then Bx

i =

   maxw

  • Bx

<i, Bx′ ≪y′ ∪ {i}

  • , if i or x ∈ S

maxw

  • Bx

<i, C x′,y′ <i

∪ {i}

  • , if i, x /

∈ S x i

Vi

If {i, x} / ∈ E: x is non-adjacent to any vertex of Vi ⇒ Ai

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 12 / 21

slide-52
SLIDE 52

Recursive formulation for Bx

i

Let x′ = ℓ- min{i, x} and let y′ = {i, x} \ x′: If {i, x} / ∈ E, then Bx

i = Ai

If {i, x} ∈ E, then Bx

i =

   maxw

  • Bx

<i, Bx′ ≪y′ ∪ {i}

  • , if i or x ∈ S

maxw

  • Bx

<i, C x′,y′ <i

∪ {i}

  • , if i, x /

∈ S x <i i

V<i

If {i, x} / ∈ E: x is non-adjacent to any vertex of Vi ⇒ Ai If {i, x} ∈ E and i / ∈ Bx

i : i is irrelevant ⇒ Bx <i

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 12 / 21

slide-53
SLIDE 53

Recursive formulation for Bx

i

Let x′ = ℓ- min{i, x} and let y′ = {i, x} \ x′: If {i, x} / ∈ E, then Bx

i = Ai

If {i, x} ∈ E, then Bx

i =

   maxw

  • Bx

<i, Bx′ ≪y′ ∪ {i}

  • , if i or x ∈ S

maxw

  • Bx

<i, C x′,y′ <i

∪ {i}

  • , if i, x /

∈ S x <i i

V<i

If {i, x} / ∈ E: x is non-adjacent to any vertex of Vi ⇒ Ai If {i, x} ∈ E and i / ∈ Bx

i : i is irrelevant ⇒ Bx <i

If {i, x} ∈ E and i ∈ Bx

i :

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 12 / 21

slide-54
SLIDE 54

Recursive formulation for Bx

i

Let x′ = ℓ- min{i, x} and let y′ = {i, x} \ x′: If {i, x} / ∈ E, then Bx

i = Ai

If {i, x} ∈ E, then Bx

i =

   maxw

  • Bx

<i, Bx′ ≪y′ ∪ {i}

  • , if i or x ∈ S

maxw

  • Bx

<i, C x′,y′ <i

∪ {i}

  • , if i, x /

∈ S x <i i ≪ x

V≪x

If {i, x} / ∈ E: x is non-adjacent to any vertex of Vi ⇒ Ai If {i, x} ∈ E and i / ∈ Bx

i : i is irrelevant ⇒ Bx <i

If {i, x} ∈ E and i ∈ Bx

i :

i or x ∈ S: every vertex between ≪x and i induces an S-triangle (Bi

≪x)

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 12 / 21

slide-55
SLIDE 55

Recursive formulation for Bx

i

Let x′ = ℓ- min{i, x} and let y′ = {i, x} \ x′: If {i, x} / ∈ E, then Bx

i = Ai

If {i, x} ∈ E, then Bx

i =

   maxw

  • Bx

<i, Bx′ ≪y′ ∪ {i}

  • , if i or x ∈ S

maxw

  • Bx

<i, C x′,y′ <i

∪ {i}

  • , if i, x /

∈ S x <i i ≪ x

V<i

If {i, x} / ∈ E: x is non-adjacent to any vertex of Vi ⇒ Ai If {i, x} ∈ E and i / ∈ Bx

i : i is irrelevant ⇒ Bx <i

If {i, x} ∈ E and i ∈ Bx

i :

i or x ∈ S: every vertex between ≪x and i induces an S-triangle (Bi

≪x)

i / ∈ S and x / ∈ S: C i,x

<i contains no S-cycle

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 12 / 21

slide-56
SLIDE 56

Recursive formulation for C x,y

i

Let x′ = ℓ- min{i, x, y} and let y′ = ℓ- min({i, x, y} \ {x′}): If {i, y} / ∈ E, then C x,y

i

= Bx

i

If {i, y} ∈ E, then C x,y

i

=

  • C x,y

<i

, if i ∈ S maxw

  • C x,y

<i , C x′,y′ <i

∪ {i}

  • , if i /

∈ S y x i

Vi

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 13 / 21

slide-57
SLIDE 57

Recursive formulation for C x,y

i

Let x′ = ℓ- min{i, x, y} and let y′ = ℓ- min({i, x, y} \ {x′}): If {i, y} / ∈ E, then C x,y

i

= Bx

i

If {i, y} ∈ E, then C x,y

i

=

  • C x,y

<i

, if i ∈ S maxw

  • C x,y

<i , C x′,y′ <i

∪ {i}

  • , if i /

∈ S y x i

Vi

If {i, y} / ∈ E: y is non-adjacent to any vertex of Vi ⇒ Bx

i

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 13 / 21

slide-58
SLIDE 58

Recursive formulation for C x,y

i

Let x′ = ℓ- min{i, x, y} and let y′ = ℓ- min({i, x, y} \ {x′}): If {i, y} / ∈ E, then C x,y

i

= Bx

i

If {i, y} ∈ E, then C x,y

i

=

  • C x,y

<i

, if i ∈ S maxw

  • C x,y

<i , C x′,y′ <i

∪ {i}

  • , if i /

∈ S y x i

Vi

If {i, y} / ∈ E: y is non-adjacent to any vertex of Vi ⇒ Bx

i

If {i, y} ∈ E:

i ∈ S: < i, x, y > is an S-triangle ⇒ i / ∈ C x,y

i

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 13 / 21

slide-59
SLIDE 59

Recursive formulation for C x,y

i

Let x′ = ℓ- min{i, x, y} and let y′ = ℓ- min({i, x, y} \ {x′}): If {i, y} / ∈ E, then C x,y

i

= Bx

i

If {i, y} ∈ E, then C x,y

i

=

  • C x,y

<i

, if i ∈ S maxw

  • C x,y

<i , C x′,y′ <i

∪ {i}

  • , if i /

∈ S y x i <i

V<i

If {i, y} / ∈ E: y is non-adjacent to any vertex of Vi ⇒ Bx

i

If {i, y} ∈ E:

i ∈ S: < i, x, y > is an S-triangle ⇒ i / ∈ C x,y

i

i / ∈ C x,y

i

: i is irrelevant ⇒ C x,y

<i

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 13 / 21

slide-60
SLIDE 60

Recursive formulation for C x,y

i

Let x′ = ℓ- min{i, x, y} and let y′ = ℓ- min({i, x, y} \ {x′}): If {i, y} / ∈ E, then C x,y

i

= Bx

i

If {i, y} ∈ E, then C x,y

i

=

  • C x,y

<i

, if i ∈ S maxw

  • C x,y

<i , C x′,y′ <i

∪ {i}

  • , if i /

∈ S y x i <i

V<i

If {i, y} / ∈ E: y is non-adjacent to any vertex of Vi ⇒ Bx

i

If {i, y} ∈ E:

i ∈ S: < i, x, y > is an S-triangle ⇒ i / ∈ C x,y

i

i / ∈ C x,y

i

: i is irrelevant ⇒ C x,y

<i

If {i, y} ∈ E and i ∈ C x,y

i

: S ∩ {i, x, y} = ∅

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 13 / 21

slide-61
SLIDE 61

Recursive formulation for C x,y

i

Let x′ = ℓ- min{i, x, y} and let y′ = ℓ- min({i, x, y} \ {x′}): If {i, y} / ∈ E, then C x,y

i

= Bx

i

If {i, y} ∈ E, then C x,y

i

=

  • C x,y

<i

, if i ∈ S maxw

  • C x,y

<i , C x′,y′ <i

∪ {i}

  • , if i /

∈ S y x i <i

V<i

If {i, y} / ∈ E: y is non-adjacent to any vertex of Vi ⇒ Bx

i

If {i, y} ∈ E:

i ∈ S: < i, x, y > is an S-triangle ⇒ i / ∈ C x,y

i

i / ∈ C x,y

i

: i is irrelevant ⇒ C x,y

<i

If {i, y} ∈ E and i ∈ C x,y

i

: S ∩ {i, x, y} = ∅

C i,x

<i ∪ {i}: none of its subset can induce an S-cycle with y

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 13 / 21

slide-62
SLIDE 62

Poly-time for SFVS on Interval Graphs

Theorem

There is a poly-time algorithm that computes SFVS of an interval graph.

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 14 / 21

slide-63
SLIDE 63

Poly-time for SFVS on Interval Graphs

Theorem

There is a poly-time algorithm that computes SFVS of an interval graph.

1 We compute the predecessors < i and ≪ i for each i in linear time

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 14 / 21

slide-64
SLIDE 64

Poly-time for SFVS on Interval Graphs

Theorem

There is a poly-time algorithm that computes SFVS of an interval graph.

1 We compute the predecessors < i and ≪ i for each i in linear time 2 Scan all intervals in an ascending order with respect to <ℓ:

Compute first Ai Then compute Bx

i and C x,y i

for every x, y such that ℓ(i) < ℓ(x) < ℓ(y)

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 14 / 21

slide-65
SLIDE 65

Poly-time for SFVS on Interval Graphs

Theorem

There is a poly-time algorithm that computes SFVS of an interval graph.

1 We compute the predecessors < i and ≪ i for each i in linear time 2 Scan all intervals in an ascending order with respect to <ℓ:

Compute first Ai Then compute Bx

i and C x,y i

for every x, y such that ℓ(i) < ℓ(x) < ℓ(y)

3 At the end, output An

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 14 / 21

slide-66
SLIDE 66

Poly-time for SFVS on Interval Graphs

Theorem

There is a poly-time algorithm that computes SFVS of an interval graph.

1 We compute the predecessors < i and ≪ i for each i in linear time 2 Scan all intervals in an ascending order with respect to <ℓ:

Compute first Ai Then compute Bx

i and C x,y i

for every x, y such that ℓ(i) < ℓ(x) < ℓ(y)

3 At the end, output An

The total running time of the algorithm is O(n3).

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 14 / 21

slide-67
SLIDE 67

Permutation Graphs

b t

a a b b c c d d e e f f g g h h D: permutation diagram with Segments between two parallel lines permutation graphs: V ↔ S such that (u, v) ∈ E iff u and v intersect Every induced cycle of a permutation graph is a triangle or a square

a b c d e f g h ✶

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 15 / 21

slide-68
SLIDE 68

Permutation Graphs

b t

a a b b c c d d e e f f g g h h D: permutation diagram with Segments between two parallel lines permutation graphs: V ↔ S such that (u, v) ∈ E iff u and v intersect Every induced cycle of a permutation graph is a triangle or a square Compute maximal S-forests FS based on D Dynamic programming approach

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 15 / 21

slide-69
SLIDE 69

Permutation Graphs

b t

a a b b c c d d e e f f g g h h D: permutation diagram with Segments between two parallel lines permutation graphs: V ↔ S such that (u, v) ∈ E iff u and v intersect Every induced cycle of a permutation graph is a triangle or a square Compute maximal S-forests FS based on D Dynamic programming approach based on crossing pairs

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 15 / 21

slide-70
SLIDE 70

Crossing Pairs

b t

a a b b c c d d e e f f g g h h X: ij with i ≤t j and j ≤b i

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 16 / 21

slide-71
SLIDE 71

Crossing Pairs

b t

a a b b c c d d e e f f g g h h X: ij with i ≤t j and j ≤b i Given gh, ij ∈ X, we define ≤ℓ and ≤r:

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 16 / 21

slide-72
SLIDE 72

Crossing Pairs

b t

a a b b c c d d e e f f g g h h X: ij with i ≤t j and j ≤b i Given gh, ij ∈ X, we define ≤ℓ and ≤r: gh ≤ℓ ij ⇔ g ≤t i and h ≤b j

b t

i i j j

b t

i i j j

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 16 / 21

slide-73
SLIDE 73

Crossing Pairs

b t

a a b b c c d d e e f f g g h h X: ij with i ≤t j and j ≤b i Given gh, ij ∈ X, we define ≤ℓ and ≤r: gh ≤ℓ ij ⇔ g ≤t i and h ≤b j

b t

i i j j

b t

i i j j

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 16 / 21

slide-74
SLIDE 74

Crossing Pairs

b t

a a b b c c d d e e f f g g h h X: ij with i ≤t j and j ≤b i Given gh, ij ∈ X, we define ≤ℓ and ≤r: gh ≤ℓ ij ⇔ g ≤t i and h ≤b j

b t

i i j j h h g g

b t

i i j j

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 16 / 21

slide-75
SLIDE 75

Crossing Pairs

b t

a a b b c c d d e e f f g g h h X: ij with i ≤t j and j ≤b i Given gh, ij ∈ X, we define ≤ℓ and ≤r: gh ≤ℓ ij ⇔ g ≤t i and h ≤b j gh ≤r ij ⇔ g ≤b i and h ≤t j

b t

i i j j h h g g

b t

i i j j

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 16 / 21

slide-76
SLIDE 76

Crossing Pairs

b t

a a b b c c d d e e f f g g h h X: ij with i ≤t j and j ≤b i Given gh, ij ∈ X, we define ≤ℓ and ≤r: gh ≤ℓ ij ⇔ g ≤t i and h ≤b j gh ≤r ij ⇔ g ≤b i and h ≤t j

b t

i i j j h h g g

b t

i i j j

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 16 / 21

slide-77
SLIDE 77

Crossing Pairs

b t

a a b b c c d d e e f f g g h h X: ij with i ≤t j and j ≤b i Given gh, ij ∈ X, we define ≤ℓ and ≤r: gh ≤ℓ ij ⇔ g ≤t i and h ≤b j gh ≤r ij ⇔ g ≤b i and h ≤t j

b t

i i j j h h g g

b t

i i j j h h g g

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 16 / 21

slide-78
SLIDE 78

Predecessors of Crossing Pairs

b t

a a b b c c d d e e f f g g h h

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 17 / 21

slide-79
SLIDE 79

Predecessors of Crossing Pairs

b t

a a b b c c d d e e f f g g h h

Veg

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 17 / 21

slide-80
SLIDE 80

Predecessors of Crossing Pairs

b t

a a b b c c d d e e f f g g h h

Veg

ij = r- max X[Vij \ {j}] the iy with y the rightmost top

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 17 / 21

slide-81
SLIDE 81

Predecessors of Crossing Pairs

b t

a a b b c c d d e e f f g g h h

Veg

ij = r- max X[Vij \ {j}] the iy with y the rightmost top ij = r- max X[Vij \ {i}] the xj with x the rightmost bottom

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 17 / 21

slide-82
SLIDE 82

Predecessors of Crossing Pairs

b t

a a b b c c d d e e f f g g h h

Veg

ij = r- max X[Vij \ {j}] the iy with y the rightmost top ij = r- max X[Vij \ {i}] the xj with x the rightmost bottom <ij = r- max X[Vij \ {i, j}] the xy with x and y the rightmost bottom and top

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 17 / 21

slide-83
SLIDE 83

Predecessors of Crossing Pairs

b t

a a b b c c d d e e f f g g h h

Veg

ij = r- max X[Vij \ {j}] the iy with y the rightmost top ij = r- max X[Vij \ {i}] the xj with x the rightmost bottom <ij = r- max X[Vij \ {i, j}] the xy with x and y the rightmost bottom and top ≪ij = r- max X[Vij \ ({i, j} ∪ {h ∈ V : {h, i} ∈ E or {h, j} ∈ E})] non-adjacent xy with x and y the rightmost bottom and top

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 17 / 21

slide-84
SLIDE 84

Basic sets: A, B, C

Sets used by our dynamic programming algorithm A: corresponds to a maximum S-forest B and C: choose a solution only from a predescribed set xy, zw ∈ V − Vij such that xy <ℓ zw and {x, w}, {y, z} ∈ E Aij = max

w {X ⊆ Vij : G[X] ∈ FS}

b t

i i j j Vij ✶ ✶

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 18 / 21

slide-85
SLIDE 85

Basic sets: A, B, C

Sets used by our dynamic programming algorithm A: corresponds to a maximum S-forest B and C: choose a solution only from a predescribed set xy, zw ∈ V − Vij such that xy <ℓ zw and {x, w}, {y, z} ∈ E Aij = max

w {X ⊆ Vij : G[X] ∈ FS}

Bxy

ij

= max

w {X ⊆ Vij : G[X ∪ {x, y}] ∈ FS}

b t

i i j j x x y y Vij ✶ ✶

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 18 / 21

slide-86
SLIDE 86

Basic sets: A, B, C

Sets used by our dynamic programming algorithm A: corresponds to a maximum S-forest B and C: choose a solution only from a predescribed set xy, zw ∈ V − Vij such that xy <ℓ zw and {x, w}, {y, z} ∈ E Aij = max

w {X ⊆ Vij : G[X] ∈ FS}

Bxy

ij

= max

w {X ⊆ Vij : G[X ∪ {x, y}] ∈ FS}

C xy,zw

ij

= max

w {X ⊆ Vij : G[X ∪ {x, y, z, w}] ∈ FS}

b t

i i j j x x y y z z w w Vij ✶

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 18 / 21

slide-87
SLIDE 87

Recursive formulations for A, B, C

Aij = maxw{X ⊆ Vij : G[X] ∈ FS}

b t

i i j j Vij ✶

Bxy

ij = maxw{X ⊆ Vij : G[X ∪ {x, y}] ∈ FS}

b t

i i j j x x y y Vij ✶

C xy,zw

ij

= maxw{X ⊆ Vij : G[X ∪ {x, y, z, w}] ∈ FS}

b t

i i j j x x y y z z w w Vij ✶

The basic idea: we support any big S-cycle with a smaller S-cycle ⇒

b t

i i j j Vij

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 19 / 21

slide-88
SLIDE 88

Recursive formulations for A, B, C

Aij = maxw{X ⊆ Vij : G[X] ∈ FS}

b t

i i j j Vij ✶

Bxy

ij = maxw{X ⊆ Vij : G[X ∪ {x, y}] ∈ FS}

b t

i i j j x x y y Vij ✶

C xy,zw

ij

= maxw{X ⊆ Vij : G[X ∪ {x, y, z, w}] ∈ FS}

b t

i i j j x x y y z z w w Vij ✶

The basic idea: we support any big S-cycle with a smaller S-cycle ⇒

b t

i i j j Vij Aij

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 19 / 21

slide-89
SLIDE 89

Recursive formulations for A, B, C

Aij = maxw{X ⊆ Vij : G[X] ∈ FS}

b t

i i j j Vij ✶

Bxy

ij = maxw{X ⊆ Vij : G[X ∪ {x, y}] ∈ FS}

b t

i i j j x x y y Vij ✶

C xy,zw

ij

= maxw{X ⊆ Vij : G[X ∪ {x, y, z, w}] ∈ FS}

b t

i i j j x x y y z z w w Vij ✶

The basic idea: we support any big S-cycle with a smaller S-cycle ⇒

b t

i i j j Vij Aij

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 19 / 21

slide-90
SLIDE 90

Recursive formulations for A, B, C

Aij = maxw{X ⊆ Vij : G[X] ∈ FS}

b t

i i j j Vij ✶

Bxy

ij = maxw{X ⊆ Vij : G[X ∪ {x, y}] ∈ FS}

b t

i i j j x x y y Vij ✶

C xy,zw

ij

= maxw{X ⊆ Vij : G[X ∪ {x, y, z, w}] ∈ FS}

b t

i i j j x x y y z z w w Vij ✶

The basic idea: we support any big S-cycle with a smaller S-cycle ⇒

b t

i i j j V<ij Bij

<ij

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 19 / 21

slide-91
SLIDE 91

Recursive formulations for A, B, C

Aij = maxw{X ⊆ Vij : G[X] ∈ FS}

b t

i i j j Vij ✶

Bxy

ij = maxw{X ⊆ Vij : G[X ∪ {x, y}] ∈ FS}

b t

i i j j x x y y Vij ✶

C xy,zw

ij

= maxw{X ⊆ Vij : G[X ∪ {x, y, z, w}] ∈ FS}

b t

i i j j x x y y z z w w Vij ✶

The basic idea: we support any big S-cycle with a smaller S-cycle ⇒

b t

i i j j V≪ij A≪ij

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 19 / 21

slide-92
SLIDE 92

Recursive formulations for A, B, C

Aij = maxw{X ⊆ Vij : G[X] ∈ FS}

b t

i i j j Vij ✶

Bxy

ij = maxw{X ⊆ Vij : G[X ∪ {x, y}] ∈ FS}

b t

i i j j x x y y Vij ✶

C xy,zw

ij

= maxw{X ⊆ Vij : G[X ∪ {x, y, z, w}] ∈ FS}

b t

i i j j x x y y z z w w Vij ✶

The basic idea: we support any big S-cycle with a smaller S-cycle ⇒

b t

i i j j V≪jj Bii

≪jj

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 19 / 21

slide-93
SLIDE 93

Recursive formulations for A, B, C

Aij = maxw{X ⊆ Vij : G[X] ∈ FS}

b t

i i j j Vij ✶

Bxy

ij = maxw{X ⊆ Vij : G[X ∪ {x, y}] ∈ FS}

b t

i i j j x x y y Vij ✶

C xy,zw

ij

= maxw{X ⊆ Vij : G[X ∪ {x, y, z, w}] ∈ FS}

b t

i i j j x x y y z z w w Vij ✶

The basic idea: we support any big S-cycle with a smaller S-cycle ⇒

b t

i i j j V≪ii Bjj

≪ii

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 19 / 21

slide-94
SLIDE 94

Poly-time for SFVS on Permutation Graphs

Theorem

There is a poly-time algorithm that computes SFVS of a permutation graph.

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 20 / 21

slide-95
SLIDE 95

Poly-time for SFVS on Permutation Graphs

Theorem

There is a poly-time algorithm that computes SFVS of a permutation graph.

1 X: there are n + m crossing pairs

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 20 / 21

slide-96
SLIDE 96

Poly-time for SFVS on Permutation Graphs

Theorem

There is a poly-time algorithm that computes SFVS of a permutation graph.

1 X: there are n + m crossing pairs 2 For each ij ∈ X, we compute its {, , <, ≪} ⇒ O(n2m)

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 20 / 21

slide-97
SLIDE 97

Poly-time for SFVS on Permutation Graphs

Theorem

There is a poly-time algorithm that computes SFVS of a permutation graph.

1 X: there are n + m crossing pairs 2 For each ij ∈ X, we compute its {, , <, ≪} ⇒ O(n2m) 3 Scan all crossing pairs in an ascending order with respect to <r:

First compute Aij Then compute Bxy

ij

for every xy ∈ V − Vij And for each xy we compute C xy,zw

ij

for every zw ∈ V − Vxy

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 20 / 21

slide-98
SLIDE 98

Poly-time for SFVS on Permutation Graphs

Theorem

There is a poly-time algorithm that computes SFVS of a permutation graph.

1 X: there are n + m crossing pairs 2 For each ij ∈ X, we compute its {, , <, ≪} ⇒ O(n2m) 3 Scan all crossing pairs in an ascending order with respect to <r:

First compute Aij Then compute Bxy

ij

for every xy ∈ V − Vij And for each xy we compute C xy,zw

ij

for every zw ∈ V − Vxy

4 At the end, output Aπ(n)n

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 20 / 21

slide-99
SLIDE 99

Poly-time for SFVS on Permutation Graphs

Theorem

There is a poly-time algorithm that computes SFVS of a permutation graph.

1 X: there are n + m crossing pairs 2 For each ij ∈ X, we compute its {, , <, ≪} ⇒ O(n2m) 3 Scan all crossing pairs in an ascending order with respect to <r:

First compute Aij Then compute Bxy

ij

for every xy ∈ V − Vij And for each xy we compute C xy,zw

ij

for every zw ∈ V − Vxy

4 At the end, output Aπ(n)n

The total running time of the algorithm is O(n + m3).

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 20 / 21

slide-100
SLIDE 100

Conclusion - Future work

AT-free

SFVS:?

chordal

SFVS:NP-c

co-bipartite

SFVS: P

permutation

SFVS: P

interval

SFVS: P

split

SFVS:NP-c FVS:P

⊃ ⊃ ⊂ ⊂ ⊃

1

Complexity on other graph classes:

⊲ AT-free graphs ⊃ co-comparability graphs, I3-free graphs ⊲ strongly chordal graphs, circular-arc graphs

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 21 / 21

slide-101
SLIDE 101

Conclusion - Future work

AT-free

SFVS:?

chordal

SFVS:NP-c

co-bipartite

SFVS: P

permutation

SFVS: P

interval

SFVS: P

split

SFVS:NP-c FVS:P

⊃ ⊃ ⊂ ⊂ ⊃

1

Complexity on other graph classes:

⊲ AT-free graphs ⊃ co-comparability graphs, I3-free graphs ⊲ strongly chordal graphs, circular-arc graphs

Graphs of bounded structural parameter

⊲ bounded clique-width: FVS ∈ P (low exp-dep.) ⇒ SFVS ∈ ? ⊲ bounded induced matching width: due to the d.p. approach

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 21 / 21

slide-102
SLIDE 102

Conclusion - Future work

AT-free

SFVS:?

chordal

SFVS:NP-c

co-bipartite

SFVS: P

permutation

SFVS: P

interval

SFVS: P

split

SFVS:NP-c FVS:P

⊃ ⊃ ⊂ ⊂ ⊃

1

Complexity on other graph classes:

⊲ AT-free graphs ⊃ co-comparability graphs, I3-free graphs ⊲ strongly chordal graphs, circular-arc graphs

Graphs of bounded structural parameter

⊲ bounded clique-width: FVS ∈ P (low exp-dep.) ⇒ SFVS ∈ ? ⊲ bounded induced matching width: due to the d.p. approach

SFVS is related to terminal-sets problems: Multiway-Cut problem: disconnect a given set of terminals Can we adopt our algorithm on permutation or interval graphs in

  • rder to work for Multiway-Cut?
  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 21 / 21

slide-103
SLIDE 103

Conclusion - Future work

AT-free

SFVS:?

chordal

SFVS:NP-c

co-bipartite

SFVS: P

permutation

SFVS: P

interval

SFVS: P

split

SFVS:NP-c FVS:P

⊃ ⊃ ⊂ ⊂ ⊃

1

Complexity on other graph classes:

⊲ AT-free graphs ⊃ co-comparability graphs, I3-free graphs ⊲ strongly chordal graphs, circular-arc graphs

Graphs of bounded structural parameter

⊲ bounded clique-width: FVS ∈ P (low exp-dep.) ⇒ SFVS ∈ ? ⊲ bounded induced matching width: due to the d.p. approach

SFVS is related to terminal-sets problems: Multiway-Cut problem: disconnect a given set of terminals Can we adopt our algorithm on permutation or interval graphs in

  • rder to work for Multiway-Cut?

... Thank You!

  • C. Papadopoulos (UoI)

SFVS on Interval and Permutation Bordeaux, FCT 2017 21 / 21