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Parameterized graph separation problems D aniel Marx Budapest - - PowerPoint PPT Presentation

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Parameterized graph separation problems D aniel Marx Budapest University of Technology and Economics dmarx@cs.bme.hu International Workshop on Parameterized and Exact Computation September 14, 2004 Bergen, Norway Parameterized graph

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Parameterized graph separation problems

D´ aniel Marx Budapest University of Technology and Economics

dmarx@cs.bme.hu

International Workshop on Parameterized and Exact Computation September 14, 2004 Bergen, Norway

Parameterized graph separation problems – p.1/10

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Terminal separation

MINIMUM TERMINAL SEPARATION

  • Given: a graph G, an integer k, and a set T of ℓ vertices (the terminals)
  • Parameter: k, ℓ
  • Find: a set S of k vertices such that S separates every two vertices of T

Note: deleting a vertex in T separates it from every other vertex. Polynomial time solvable for ℓ = 2 (network flows). NP-hard for every ℓ = 3 [Cunningham, 1991]

Parameterized graph separation problems – p.2/10

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Terminal separation

MINIMUM TERMINAL SEPARATION

  • Given: a graph G, an integer k, and a set T of ℓ vertices (the terminals)
  • Parameter: k, ℓ
  • Find: a set S of k vertices such that S separates every two vertices of T

Note: deleting a vertex in T separates it from every other vertex. Polynomial time solvable for ℓ = 2 (network flows). NP-hard for every ℓ = 3 [Cunningham, 1991] Theorem: MINIMUM TERMINAL SEPARATION is fixed-parameter tractable with parameter k. (Follows from graph minors theory, but here we give a direct proof.)

Parameterized graph separation problems – p.2/10

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Main idea

We try to separate t1 from the rest of the terminals: t1

Parameterized graph separation problems – p.3/10

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Main idea

We try to separate t1 from the rest of the terminals: t1

Parameterized graph separation problems – p.3/10

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SLIDE 6

Main idea

We try to separate t1 from the rest of the terminals: t1 To separate t1, we try to delete vertices as far away from t1 as possible.

Parameterized graph separation problems – p.3/10

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Main idea

We try to separate t1 from the rest of the terminals: t1 To separate t1, we try to delete vertices as far away from t1 as possible.

Parameterized graph separation problems – p.3/10

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Important separators

Definition: S is an (X,Y)-separator if it separates every vertex of X from every vertex of Y . Definition: an (X, Y )-separator R dominates an (X, Y )-separator S if

  • |R| ≤ |S|,
  • every vertex reachable from X in G \ S is reachable from X in G \ R.

Definition: S is an important (X, Y )-separator, if it is not dominated by any

  • ther (X, Y )-separator.

Parameterized graph separation problems – p.4/10

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Important separators

Definition: S is an (X,Y)-separator if it separates every vertex of X from every vertex of Y . Definition: an (X, Y )-separator R dominates an (X, Y )-separator S if

  • |R| ≤ |S|,
  • every vertex reachable from X in G \ S is reachable from X in G \ R.

Definition: S is an important (X, Y )-separator, if it is not dominated by any

  • ther (X, Y )-separator.

Y j g e d c b a y4 y3 y2 y1 x1 x2 f i k h

Parameterized graph separation problems – p.4/10

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Important separators

Definition: S is an (X,Y)-separator if it separates every vertex of X from every vertex of Y . Definition: an (X, Y )-separator R dominates an (X, Y )-separator S if

  • |R| ≤ |S|,
  • every vertex reachable from X in G \ S is reachable from X in G \ R.

Definition: S is an important (X, Y )-separator, if it is not dominated by any

  • ther (X, Y )-separator.

Y j g e d c b a y4 y3 y2 y1 x1 x2 f i k h

Parameterized graph separation problems – p.4/10

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SLIDE 11

Important separators

Definition: S is an (X,Y)-separator if it separates every vertex of X from every vertex of Y . Definition: an (X, Y )-separator R dominates an (X, Y )-separator S if

  • |R| ≤ |S|,
  • every vertex reachable from X in G \ S is reachable from X in G \ R.

Definition: S is an important (X, Y )-separator, if it is not dominated by any

  • ther (X, Y )-separator.

Y j g e d c b a y4 y3 y2 y1 x1 x2 f i k h

Separator {b, f} is dominated by {c, i}.

Parameterized graph separation problems – p.4/10

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Important separators

Definition: S is an (X,Y)-separator if it separates every vertex of X from every vertex of Y . Definition: an (X, Y )-separator R dominates an (X, Y )-separator S if

  • |R| ≤ |S|,
  • every vertex reachable from X in G \ S is reachable from X in G \ R.

Definition: S is an important (X, Y )-separator, if it is not dominated by any

  • ther (X, Y )-separator.

Y j g e d c b a y4 y3 y2 y1 x1 x2 f i k h

Important separators: {c, i}

Parameterized graph separation problems – p.4/10

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SLIDE 13

Important separators

Definition: S is an (X,Y)-separator if it separates every vertex of X from every vertex of Y . Definition: an (X, Y )-separator R dominates an (X, Y )-separator S if

  • |R| ≤ |S|,
  • every vertex reachable from X in G \ S is reachable from X in G \ R.

Definition: S is an important (X, Y )-separator, if it is not dominated by any

  • ther (X, Y )-separator.

Y j g e d c b a y4 y3 y2 y1 x1 x2 f i k h

Important separators: {c, i} {d, e, i}

Parameterized graph separation problems – p.4/10

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SLIDE 14

Important separators

Definition: S is an (X,Y)-separator if it separates every vertex of X from every vertex of Y . Definition: an (X, Y )-separator R dominates an (X, Y )-separator S if

  • |R| ≤ |S|,
  • every vertex reachable from X in G \ S is reachable from X in G \ R.

Definition: S is an important (X, Y )-separator, if it is not dominated by any

  • ther (X, Y )-separator.

Y j g e d c b a y4 y3 y2 y1 x1 x2 f i k h

Important separators: {c, i} {d, e, i} {c, j, k}

Parameterized graph separation problems – p.4/10

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Important separators

Definition: S is an (X,Y)-separator if it separates every vertex of X from every vertex of Y . Definition: an (X, Y )-separator R dominates an (X, Y )-separator S if

  • |R| ≤ |S|,
  • every vertex reachable from X in G \ S is reachable from X in G \ R.

Definition: S is an important (X, Y )-separator, if it is not dominated by any

  • ther (X, Y )-separator.

Y j g e d c b a y4 y3 y2 y1 x1 x2 f i k h

Important separators: {c, i} {d, e, i} {c, j, k} {y1, y2, y3, y4}

Parameterized graph separation problems – p.4/10

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Number of important separators

We want to bound the number of important separators. Question: What is the maximum number of size 1 important (X, Y )-separators?

Parameterized graph separation problems – p.5/10

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Number of important separators

We want to bound the number of important separators. Question: What is the maximum number of size 1 important (X, Y )-separators? Answer: 1

Parameterized graph separation problems – p.5/10

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Number of important separators

We want to bound the number of important separators. Question: What is the maximum number of size 1 important (X, Y )-separators? Answer: 1 X Y

Parameterized graph separation problems – p.5/10

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Number of important separators

We want to bound the number of important separators. Question: What is the maximum number of size 1 important (X, Y )-separators? Answer: 1 X Y

Parameterized graph separation problems – p.5/10

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Number of important separators

We want to bound the number of important separators. Question: What is the maximum number of size 1 important (X, Y )-separators? Answer: 1 X Y The unique important separator

Parameterized graph separation problems – p.5/10

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Number

  • f

important separators (cont.)

Another example: ct c2 b2 Y c1 at . . . . . . a2 a1 X b1 bt bt For every i, the separator has to contain either

  • ai or
  • both bi and ci.

Parameterized graph separation problems – p.6/10

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Number

  • f

important separators (cont.)

Another example: ct c2 b2 Y c1 at . . . . . . a2 a1 X b1 bt bt For every i, the separator has to contain either

  • ai or
  • both bi and ci.

Parameterized graph separation problems – p.6/10

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SLIDE 23

Number

  • f

important separators (cont.)

Another example: ct c2 b2 Y c1 at . . . . . . a2 a1 X b1 bt bt For every i, the separator has to contain either

  • ai or
  • both bi and ci.

Parameterized graph separation problems – p.6/10

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SLIDE 24

Number

  • f

important separators (cont.)

Another example: ct c2 b2 Y c1 at . . . . . . a2 a1 X b1 bt bt For every i, the separator has to contain either

  • ai or
  • both bi and ci.

Every combination gives an im- portant separator ⇒ there are 2t important separators of size at most 2t.

Parameterized graph separation problems – p.6/10

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The two key lemmas

Lemma 1: There are at most 4k2 important (X, Y )-separators of size ≤ k. Lemma 2: If the terminals t1, t2, . . . , tℓ can be separated by deleting k vertices, then there is a solution that contains an important ({t1}, {t2, . . . , tℓ})-separator.

Parameterized graph separation problems – p.7/10

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The two key lemmas

Lemma 1: There are at most 4k2 important (X, Y )-separators of size ≤ k. Lemma 2: If the terminals t1, t2, . . . , tℓ can be separated by deleting k vertices, then there is a solution that contains an important ({t1}, {t2, . . . , tℓ})-separator. Algorithm:

  • 1. Enumerate all the important ({t1}, {t2, . . . , tℓ})-separators of size at

most k.

  • 2. Delete one of them from the graph.
  • 3. Decrease the parameter k, and go to Step 1.

Bounded search tree: branch factor is at most 4k2, height is at most k.

Parameterized graph separation problems – p.7/10

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Proof of Lemma 1

Lemma 1: There are at most 4k2 important (X, Y )-separators of size ≤ k. Proof: By induction on k, we have seen the case k = 1. Let S be an important separator. How many other important separators are possible?

Parameterized graph separation problems – p.8/10

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Proof of Lemma 1

Lemma 1: There are at most 4k2 important (X, Y )-separators of size ≤ k. Proof: By induction on k, we have seen the case k = 1. Let S be an important separator. How many other important separators are possible?

X Y

Parameterized graph separation problems – p.8/10

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Proof of Lemma 1

Lemma 1: There are at most 4k2 important (X, Y )-separators of size ≤ k. Proof: By induction on k, we have seen the case k = 1. Let S be an important separator. How many other important separators are possible?

X Y S

Parameterized graph separation problems – p.8/10

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Proof of Lemma 1

Lemma 1: There are at most 4k2 important (X, Y )-separators of size ≤ k. Proof: By induction on k, we have seen the case k = 1. Let S be an important separator. How many other important separators are possible?

X Y S R1 R2

Let R = R1 ∪ R2 be an important separator disjoint from S.

Parameterized graph separation problems – p.8/10

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Proof of Lemma 1

Lemma 1: There are at most 4k2 important (X, Y )-separators of size ≤ k. Proof: By induction on k, we have seen the case k = 1. Let S be an important separator. How many other important separators are possible?

X Y S R1 R2 k2 k1

Let R = R1 ∪ R2 be an important separator disjoint from S.

Parameterized graph separation problems – p.8/10

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Proof of Lemma 1

Lemma 1: There are at most 4k2 important (X, Y )-separators of size ≤ k. Proof: By induction on k, we have seen the case k = 1. Let S be an important separator. How many other important separators are possible?

X Y S R

Let R = R1 ∪ R2 be an important separator disjoint from S.

  • If k1 = k ⇒ R is not important.

Parameterized graph separation problems – p.8/10

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Proof of Lemma 1

Lemma 1: There are at most 4k2 important (X, Y )-separators of size ≤ k. Proof: By induction on k, we have seen the case k = 1. Let S be an important separator. How many other important separators are possible?

X Y S R

Let R = R1 ∪ R2 be an important separator disjoint from S.

  • If k1 = k ⇒ R is not important.
  • If k2 = k ⇒ S is not important.

Parameterized graph separation problems – p.8/10

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Proof of Lemma 1

Lemma 1: There are at most 4k2 important (X, Y )-separators of size ≤ k. Proof: By induction on k, we have seen the case k = 1. Let S be an important separator. How many other important separators are possible?

X Y S R1 R2 k2 k1

Let R = R1 ∪ R2 be an important separator disjoint from S.

  • If k1 = k ⇒ R is not important.
  • If k2 = k ⇒ S is not important.

It can be shown that R1 (resp. R2) is an important (X′, Y ′)-separator (for some X′, Y ′) ⇒ constant number of possibilities for R1 and R2 ⇒ constant number of possibilities for R.

Parameterized graph separation problems – p.8/10

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Proof of Lemma 2

Lemma 2: If the terminals t1, t2, . . . , tℓ can be separated by deleting k vertices, then there is a solution that contains an important ({t1}, {t2, . . . , tℓ})-separator.

Parameterized graph separation problems – p.9/10

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Proof of Lemma 2

Lemma 2: If the terminals t1, t2, . . . , tℓ can be separated by deleting k vertices, then there is a solution that contains an important ({t1}, {t2, . . . , tℓ})-separator. t2 t1 t3 t5 t6 t4 S

Parameterized graph separation problems – p.9/10

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Proof of Lemma 2

Lemma 2: If the terminals t1, t2, . . . , tℓ can be separated by deleting k vertices, then there is a solution that contains an important ({t1}, {t2, . . . , tℓ})-separator. t2 t1 t3 t5 t6 t4 S S1

  • Let S1 ⊆ S be those vertices that can be

reached from t1 without entering S.

  • If S1 is not an important separator, then it

is dominated by some S2. We show that in this case S′ = (S \ S1) ∪ S2 separates the terminals.

Parameterized graph separation problems – p.9/10

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Proof of Lemma 2

Lemma 2: If the terminals t1, t2, . . . , tℓ can be separated by deleting k vertices, then there is a solution that contains an important ({t1}, {t2, . . . , tℓ})-separator. t2 t1 t3 t5 t6 t4 S S1

  • Let S1 ⊆ S be those vertices that can be

reached from t1 without entering S.

  • If S1 is not an important separator, then it

is dominated by some S2. We show that in this case S′ = (S \ S1) ∪ S2 separates the terminals.

  • t1 is separated from every terminal by S2.

Parameterized graph separation problems – p.9/10

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Proof of Lemma 2

Lemma 2: If the terminals t1, t2, . . . , tℓ can be separated by deleting k vertices, then there is a solution that contains an important ({t1}, {t2, . . . , tℓ})-separator. t2 t1 t3 t5 t6 t4 S S1

  • Let S1 ⊆ S be those vertices that can be

reached from t1 without entering S.

  • If S1 is not an important separator, then it

is dominated by some S2. We show that in this case S′ = (S \ S1) ∪ S2 separates the terminals.

  • t1 is separated from every terminal by S2.

Parameterized graph separation problems – p.9/10

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Proof of Lemma 2

Lemma 2: If the terminals t1, t2, . . . , tℓ can be separated by deleting k vertices, then there is a solution that contains an important ({t1}, {t2, . . . , tℓ})-separator. t2 t1 t3 t5 t6 t4 S S1 v

  • Let S1 ⊆ S be those vertices that can be

reached from t1 without entering S.

  • If S1 is not an important separator, then it

is dominated by some S2. We show that in this case S′ = (S \ S1) ∪ S2 separates the terminals.

  • t1 is separated from every terminal by S2.
  • If t2 and t3 are connected, then the path

has to go through some v ∈ S1

Parameterized graph separation problems – p.9/10

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Proof of Lemma 2

Lemma 2: If the terminals t1, t2, . . . , tℓ can be separated by deleting k vertices, then there is a solution that contains an important ({t1}, {t2, . . . , tℓ})-separator. t2 t1 t3 t5 t6 t4 S S1 v

  • Let S1 ⊆ S be those vertices that can be

reached from t1 without entering S.

  • If S1 is not an important separator, then it

is dominated by some S2. We show that in this case S′ = (S \ S1) ∪ S2 separates the terminals.

  • t1 is separated from every terminal by S2.
  • If t2 and t3 are connected, then the path

has to go through some v ∈ S1 ⇒ t1 is con- nected to both t2 and t3, a contradiction.

Parameterized graph separation problems – p.9/10

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Terminal pairs

MINIMUM TERMINAL PAIR SEPARATION

  • Given: a graph G, an integer k, and ℓ pairs of terminals (s1, t1), . . . ,

(sℓ, tℓ).

  • Parameter: k, ℓ
  • Find: a set S of k vertices such after deleting S, terminals si and ti are

separated for every 1 ≤ i ≤ ℓ.

Parameterized graph separation problems – p.10/10

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Terminal pairs

MINIMUM TERMINAL PAIR SEPARATION

  • Given: a graph G, an integer k, and ℓ pairs of terminals (s1, t1), . . . ,

(sℓ, tℓ).

  • Parameter: k, ℓ
  • Find: a set S of k vertices such after deleting S, terminals si and ti are

separated for every 1 ≤ i ≤ ℓ. Theorem: MINIMUM TERMINAL PAIR SEPARATION is fixed-parameter tractable with parameters k and ℓ.

Parameterized graph separation problems – p.10/10

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SLIDE 44

Terminal pairs

MINIMUM TERMINAL PAIR SEPARATION

  • Given: a graph G, an integer k, and ℓ pairs of terminals (s1, t1), . . . ,

(sℓ, tℓ).

  • Parameter: k, ℓ
  • Find: a set S of k vertices such after deleting S, terminals si and ti are

separated for every 1 ≤ i ≤ ℓ. Theorem: MINIMUM TERMINAL PAIR SEPARATION is fixed-parameter tractable with parameters k and ℓ. Algorithm: s1 is separated from t1, and from a subset X of {s2, t2, . . . , sℓ, tℓ}. Make a guess for the set X, and separate s1 with an important (s1, X ∪ t1)-separator.

Parameterized graph separation problems – p.10/10

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Terminal pairs

MINIMUM TERMINAL PAIR SEPARATION

  • Given: a graph G, an integer k, and ℓ pairs of terminals (s1, t1), . . . ,

(sℓ, tℓ).

  • Parameter: k, ℓ
  • Find: a set S of k vertices such after deleting S, terminals si and ti are

separated for every 1 ≤ i ≤ ℓ. Theorem: MINIMUM TERMINAL PAIR SEPARATION is fixed-parameter tractable with parameters k and ℓ. Algorithm: s1 is separated from t1, and from a subset X of {s2, t2, . . . , sℓ, tℓ}. Make a guess for the set X, and separate s1 with an important (s1, X ∪ t1)-separator. Open: What is the complexity of the problem if only k is the parameter?

Parameterized graph separation problems – p.10/10