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

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

Terminal separation M INIMUM T ERMINAL S EPARATION • 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

Terminal separation M INIMUM T ERMINAL S EPARATION • 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: M INIMUM T ERMINAL S EPARATION 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

Main idea We try to separate t 1 from the rest of the terminals: t 1 Parameterized graph separation problems – p.3/10

Main idea We try to separate t 1 from the rest of the terminals: t 1 Parameterized graph separation problems – p.3/10

Main idea We try to separate t 1 from the rest of the terminals: t 1 To separate t 1 , we try to delete vertices as far away from t 1 as possible. Parameterized graph separation problems – p.3/10

Main idea We try to separate t 1 from the rest of the terminals: t 1 To separate t 1 , we try to delete vertices as far away from t 1 as possible. Parameterized graph separation problems – p.3/10

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 other ( X, Y ) -separator. Parameterized graph separation problems – p.4/10

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 other ( X, Y ) -separator. Y d y 1 a b c y 2 x 1 e f g j i x 2 y 3 y 4 h k Parameterized graph separation problems – p.4/10

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 other ( X, Y ) -separator. Y d y 1 a b c y 2 x 1 e f g j i x 2 y 3 y 4 h k Parameterized graph separation problems – p.4/10

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 other ( X, Y ) -separator. Y d Separator { b, f } is dominated y 1 by { c, i } . a b c y 2 x 1 e f g j i x 2 y 3 y 4 h k Parameterized graph separation problems – p.4/10

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 other ( X, Y ) -separator. Y d Important separators: y 1 { c, i } a b c y 2 x 1 e f g j i x 2 y 3 y 4 h k Parameterized graph separation problems – p.4/10

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 other ( X, Y ) -separator. Y d Important separators: y 1 { c, i } a b c { d, e, i } y 2 x 1 e f g j i x 2 y 3 y 4 h k Parameterized graph separation problems – p.4/10

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 other ( X, Y ) -separator. Y d Important separators: y 1 { c, i } a b c { d, e, i } y 2 x 1 e { c, j, k } f g j i x 2 y 3 y 4 h k Parameterized graph separation problems – p.4/10

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 other ( X, Y ) -separator. Y d Important separators: y 1 { c, i } a b c { d, e, i } y 2 x 1 e { c, j, k } f g j i { y 1 , y 2 , y 3 , y 4 } x 2 y 3 y 4 h k Parameterized graph separation problems – p.4/10

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

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

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

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

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 The unique important separator X Y Parameterized graph separation problems – p.5/10

Number of important separators (cont.) Another example: For every i , the separator has to contain either Y • a i or b 1 c 1 b 2 c 2 b t b t c t . . . • both b i and c i . . . . a 1 a 2 a t X Parameterized graph separation problems – p.6/10

Number of important separators (cont.) Another example: For every i , the separator has to contain either Y • a i or b 1 c 1 b 2 c 2 b t b t c t . . . • both b i and c i . . . . a 1 a 2 a t X Parameterized graph separation problems – p.6/10

Number of important separators (cont.) Another example: For every i , the separator has to contain either Y • a i or b 1 c 1 b 2 c 2 b t b t c t . . . • both b i and c i . . . . a 1 a 2 a t X Parameterized graph separation problems – p.6/10

Number of important separators (cont.) Another example: For every i , the separator has to contain either Y • a i or b 1 c 1 b 2 c 2 b t b t c t . . . • both b i and c i . Every combination gives an im- portant separator ⇒ there are 2 t important separators of size . . . a 1 a 2 a t at most 2 t . X Parameterized graph separation problems – p.6/10

The two key lemmas Lemma 1: There are at most 4 k 2 important ( X, Y ) -separators of size ≤ k . Lemma 2: If the terminals t 1 , t 2 , . . . , t ℓ can be separated by deleting k vertices, then there is a solution that contains an important ( { t 1 } , { t 2 , . . . , t ℓ } ) -separator. Parameterized graph separation problems – p.7/10

The two key lemmas Lemma 1: There are at most 4 k 2 important ( X, Y ) -separators of size ≤ k . Lemma 2: If the terminals t 1 , t 2 , . . . , t ℓ can be separated by deleting k vertices, then there is a solution that contains an important ( { t 1 } , { t 2 , . . . , t ℓ } ) -separator. Algorithm: 1. Enumerate all the important ( { t 1 } , { t 2 , . . . , t ℓ } ) -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 4 k 2 , height is at most k . Parameterized graph separation problems – p.7/10