Important separators and parameterized algorithms Dániel Marx Humboldt-Universität zu Berlin, Germany Methods for Discrete Structures February 7, 2011 Important separators and parameterized algorithms – p.1/27
Overview Main message: Small separators in graphs have interesting extremal properties that can be exploited in combinatorial and algorithmic results. Bounding the number of “important” separators. Combinatorial application: Erd˝ os-Pósa property for “spiders.” Algorithmic applications: FPT algorithm for multiway cut and a directed feedback vertex set. Important separators and parameterized algorithms – p.2/27
Important separators Definition: δ ( R ) is the set of edges with exactly one endpoint in R . Definition: A set S of edges is an ( X , Y ) -separator if there is no X − Y path in G \ S and no proper subset of S breaks every X − Y path. Observation: Every ( X , Y ) -separator S can be expressed as S = δ ( R ) for some X ⊆ R and R ∩ Y = ∅ . δ ( R ) Y X R Important separators and parameterized algorithms – p.3/27
Important separators An ( X , Y ) -separator δ ( R ) is important if there is no ( X , Y ) - Definition: separator δ ( R ′ ) with R ⊂ R ′ and | δ ( R ′ ) | ≤ | δ ( R ) | . Note: Can be checked in polynomial time if a separator is important. δ ( R ) Y X R Important separators and parameterized algorithms – p.3/27
Important separators An ( X , Y ) -separator δ ( R ) is important if there is no ( X , Y ) - Definition: separator δ ( R ′ ) with R ⊂ R ′ and | δ ( R ′ ) | ≤ | δ ( R ) | . Note: Can be checked in polynomial time if a separator is important. δ ( R ) Y X δ ( R ′ ) R R ′ Important separators and parameterized algorithms – p.3/27
Important separators An ( X , Y ) -separator δ ( R ) is important if there is no ( X , Y ) - Definition: separator δ ( R ′ ) with R ⊂ R ′ and | δ ( R ′ ) | ≤ | δ ( R ) | . Note: Can be checked in polynomial time if a separator is important. δ ( R ) Y X R Important separators and parameterized algorithms – p.3/27
Important separators The number of important separators can be exponentially large. Example: Y k / 2 1 2 X This graph has exactly 2 k / 2 important ( X , Y ) -separators of size at most k . Theorem: There are at most 4 k important ( X , Y ) -separators of size at most k . (Proof is implicit in [Chen, Liu, Lu 2007], worse bound in [M. 2004].) Important separators and parameterized algorithms – p.4/27
Submodularity Fact: The function δ is submodular: for arbitrary sets A , B , | δ ( A ) | + | δ ( B ) | ≥ | δ ( A ∩ B ) | + | δ ( A ∪ B ) | Consequence: Let λ be the minimum ( X , Y ) -separator size. There is a unique maximal R max ⊇ X such that δ ( R max ) is an ( X , Y ) -separator of size λ . Important separators and parameterized algorithms – p.5/27
Submodularity Fact: The function δ is submodular: for arbitrary sets A , B , | δ ( A ) | + | δ ( B ) | ≥ | δ ( A ∩ B ) | + | δ ( A ∪ B ) | Consequence: Let λ be the minimum ( X , Y ) -separator size. There is a unique maximal R max ⊇ X such that δ ( R max ) is an ( X , Y ) -separator of size λ . Proof: Let R 1 , R 2 ⊇ X be two sets such that δ ( R 1 ), δ ( R 2 ) are ( X , Y ) -separators of size λ . Y | δ ( R 1 ) | + | δ ( R 2 ) | ≥ | δ ( R 1 ∩ R 2 ) | + | δ ( R 1 ∪ R 2 ) | λ λ ≥ λ R 1 R 2 ⇒ | δ ( R 1 ∪ R 2 ) | ≤ λ X Note: Analogous result holds for a unique minimal R min . Important separators and parameterized algorithms – p.5/27
Important separators Theorem: There are at most 4 k important ( X , Y ) -separators of size at most k . Proof: Let λ be the minimum ( X , Y ) -separator size and let δ ( R max ) be the unique important separator of size λ such that R max is maximal. First we show that R max ⊆ R for every important separator δ ( R ) . Important separators and parameterized algorithms – p.6/27
Important separators Theorem: There are at most 4 k important ( X , Y ) -separators of size at most k . Proof: Let λ be the minimum ( X , Y ) -separator size and let δ ( R max ) be the unique important separator of size λ such that R max is maximal. First we show that R max ⊆ R for every important separator δ ( R ) . By the submodularity of δ : | δ ( R max ) | + | δ ( R ) | ≥ | δ ( R max ∩ R ) | + | δ ( R max ∪ R ) | ≥ λ λ ⇓ | δ ( R max ∪ R ) | ≤ | δ ( R ) | ⇓ If R � = R max ∪ R , then δ ( R ) is not important. Thus the important ( X , Y ) - and ( R max , Y ) -separators are the same. ⇒ We can assume X = R max . Important separators and parameterized algorithms – p.6/27
Important separators Lemma: There are at most 4 k important ( X , Y ) -separators of size at most k . Search tree algorithm for enumerating all these separators: An (arbitrary) edge uv leaving X = R max is either in the separator or not. Branch 1: If uv ∈ S , then S \ uv is an important ( X , Y ) -separator of size at most k − 1 in G \ uv . u v X = R max Y Branch 2: If uv �∈ S , then S is an important ( X ∪ v , Y ) -separator of size at most k in G . Important separators and parameterized algorithms – p.7/27
Important separators Lemma: There are at most 4 k important ( X , Y ) -separators of size at most k . Search tree algorithm for enumerating all these separators: An (arbitrary) edge uv leaving X = R max is either in the separator or not. Branch 1: If uv ∈ S , then S \ uv is an important ( X , Y ) -separator of size at most k − 1 in G \ uv . ⇒ k decreases by one, λ decreases by at most 1 . u v X = R max Y Branch 2: If uv �∈ S , then S is an important ( X ∪ v , Y ) -separator of size at most k in G . ⇒ k remains the same, λ increases by 1 . The measure 2 k − λ decreases in each step. ⇒ Height of the search tree ≤ 2 k ⇒ ≤ 2 2 k important separators of size ≤ k . Important separators and parameterized algorithms – p.7/27
Important separators Example: The bound 4 k is essentially tight. X Y Important separators and parameterized algorithms – p.8/27
Important separators Example: The bound 4 k is essentially tight. X Y Any subtree with k leaves gives an important ( X , Y ) -separator of size k . Important separators and parameterized algorithms – p.8/27
Important separators Example: The bound 4 k is essentially tight. X Y Any subtree with k leaves gives an important ( X , Y ) -separator of size k . Important separators and parameterized algorithms – p.8/27
Important separators Example: The bound 4 k is essentially tight. X Y Any subtree with k leaves gives an important ( X , Y ) -separator of size k . The number of subtrees with k leaves is the Catalan number � � 2 k − 2 C k − 1 = 1 ≥ 4 k / poly ( k ). k − 1 k Important separators and parameterized algorithms – p.8/27
Simple application Lemma: At most k · 4 k edges incident to t can be part of an inclusionwise minimal s − t cut of size at most k . Important separators and parameterized algorithms – p.9/27
Simple application Lemma: At most k · 4 k edges incident to t can be part of an inclusionwise minimal s − t cut of size at most k . Proof: We show that every such edge is contained in an important ( s , t ) -separator of size at most k . v s t R Suppose that vt ∈ δ ( R ) and | δ ( R ) | = k . Important separators and parameterized algorithms – p.9/27
Simple application Lemma: At most k · 4 k edges incident to t can be part of an inclusionwise minimal s − t cut of size at most k . Proof: We show that every such edge is contained in an important ( s , t ) -separator of size at most k . v s t R R ′ Suppose that vt ∈ δ ( R ) and | δ ( R ) | = k . There is an important ( s , t ) -separator δ ( R ′ ) with R ⊆ R ′ and | δ ( R ′ ) | ≤ k . Clearly, vt ∈ δ ( R ′ ) : v ∈ R , hence v ∈ R ′ . Important separators and parameterized algorithms – p.9/27
Anti isolation Let s , t 1 , ... , t n be vertices and S 1 , ... , S n be sets of at most k edges such that S i separates t i from s , but S i does not separate t j from s for any j � = i . It is possible that n is “large” even if k is “small.” t 1 t 2 t 3 t 4 t 5 t 6 s Important separators and parameterized algorithms – p.10/27
Anti isolation Let s , t 1 , ... , t n be vertices and S 1 , ... , S n be sets of at most k edges such that S i separates t i from s , but S i does not separate t j from s for any j � = i . It is possible that n is “large” even if k is “small.” t 1 t 2 t 3 t 4 t 5 t 6 S 1 s Important separators and parameterized algorithms – p.10/27
Anti isolation Let s , t 1 , ... , t n be vertices and S 1 , ... , S n be sets of at most k edges such that S i separates t i from s , but S i does not separate t j from s for any j � = i . It is possible that n is “large” even if k is “small.” t 1 t 2 t 3 t 4 t 5 t 6 S 2 s Important separators and parameterized algorithms – p.10/27
Anti isolation Let s , t 1 , ... , t n be vertices and S 1 , ... , S n be sets of at most k edges such that S i separates t i from s , but S i does not separate t j from s for any j � = i . It is possible that n is “large” even if k is “small.” t 1 t 2 t 3 t 4 t 5 t 6 S 3 s Important separators and parameterized algorithms – p.10/27
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