Information about Important Cuts

Published on March 5, 2014

Author: ASPAK2014

Source: slideshare.net

Overview Main message: Small cuts/separators in graphs have interesting extremal properties that can be exploited in combinatorial and algorithmic results. Bounding the number of “important” cuts/separators. Edge/vertex versions, directed/undirected versions. Algorithmic applications: FPT algorithm for M ULTIWAY F EEDBACK V ERTEX S ET. CUT and D IRECTED I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 2/33

Cuts An (X , Y )-cut is a set S of edges that separate X and Y for each other, that is, G S has no X − Y path. An (X , Y )-cut S is a minimum (X , Y )-cut if there is no (X , Y )-cut S ′ with |S ′ | < |S|. An (X , Y )-cut is (inclusionwise) minimal if there is no (X , Y )-cut S ′ with S′ ⊂ S I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 3/33

Characterizing Cuts Want to view them as edges on the boundary of a certain set of vertices. If G is an undirected graph and R ⊆ V (G ) is a set of vertices, then we denote by ∆G (R) the set of edges with exactly one endpoint in R . Let S be a minimal (X , Y )-cut in G and let R be the set of vertices reachable from X in G S; clearly, we have X ⊆ R ⊆ V (G ) Y . Then it is easy to see that S is precisely ∆(R). I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 4/33

Characterizing Cuts If S is a minimal (X , Y )-cut, then S = ∆(R), where R is the set of vertices reachable from X in G S. Proof on board. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 5/33

Menger’s Theorem The well-know Max-Flow Min-Cut duality implies that the size of the minimum (X , Y )-cut is the same as the maximum number of pairwise edge-disjoint X − Y paths. ALGORITHM Given a graph G , disjoint sets X , Y ⊆ V (G ), and an integer k, there is an O(k(|V (G )| + |E (G )|)) time algorithm that either correctly concludes that there is no (X , Y )-cut of size at most k, or returns a minimum (X , Y )-cut ∆(R) and a collection of |∆(R)| pairwise edge-disjoint X − Y paths. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 6/33

Submodular We say that f is submodular if it satisﬁes the following inequality for every X , Y ⊆ V (G ): f (A) + f (B) ≥ f (A ∩ B) + f (A ∪ B). (1) Let δG (or δ) denote the function from 2V (G ) → N such that δ(X ) is the number of edges with one endpoint in X and other in V (G ) X . I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 7/33

Submodularity Fact: The function δ is submodular: for arbitrary sets A, B, |δ(A)| + |δ(B)| ≥ |δ(A ∩ B)| + |δ(A ∪ B)| I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 8/33

Submodularity Fact: The function δ is submodular: for arbitrary sets A, B, |δ(A)| + |δ(B)| ≥ |δ(A ∩ B)| + |δ(A ∪ B)| Proof: Determine separately the contribution of the different types of edges. A B I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 8/33

Submodularity Fact: The function δ is submodular: for arbitrary sets A, B, |δ(A)| + |δ(B)| 0 ≥ |δ(A ∩ B)| 1 1 + |δ(A ∪ B)| 0 Proof: Determine separately the contribution of the different types of edges. A B I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 8/33

Submodularity Fact: The function δ is submodular: for arbitrary sets A, B, |δ(A)| + |δ(B)| 1 ≥ |δ(A ∩ B)| 0 1 + |δ(A ∪ B)| 0 Proof: Determine separately the contribution of the different types of edges. A B I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 8/33

Submodularity Fact: The function δ is submodular: for arbitrary sets A, B, |δ(A)| + |δ(B)| 0 ≥ |δ(A ∩ B)| 1 0 + |δ(A ∪ B)| 1 Proof: Determine separately the contribution of the different types of edges. A B I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 8/33

Submodularity Fact: The function δ is submodular: for arbitrary sets A, B, |δ(A)| + |δ(B)| 1 ≥ |δ(A ∩ B)| 0 0 + |δ(A ∪ B)| 1 Proof: Determine separately the contribution of the different types of edges. A B I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 8/33

Submodularity Fact: The function δ is submodular: for arbitrary sets A, B, |δ(A)| + |δ(B)| 1 ≥ |δ(A ∩ B)| 1 1 + |δ(A ∪ B)| 1 Proof: Determine separately the contribution of the different types of edges. A B I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 8/33

Submodularity Fact: The function δ is submodular: for arbitrary sets A, B, |δ(A)| + |δ(B)| 1 ≥ |δ(A ∩ B)| 1 0 + |δ(A ∪ B)| 0 Proof: Determine separately the contribution of the different types of edges. A B I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 8/33

Submodularity Consequence: Let λ be the minimum (X , Y )-cut size. There is a unique maximal Rmax ⊇ X such that δ(Rmax ) is an (X , Y )-cut of size λ. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 9/33

Submodularity Consequence: Let λ be the minimum (X , Y )-cut size. There is a unique maximal Rmax ⊇ X such that δ(Rmax ) is an (X , Y )-cut of size λ. Proof: Let R1 , R2 ⊇ X be two sets such that δ(R1 ), δ(R2 ) are (X , Y )-cuts of size λ. Y |δ(R1 )| + |δ(R2 )| ≥ |δ(R1 ∩ R2 )| + |δ(R1 ∪ R2 )| λ λ ≥λ R1 ⇒ |δ(R1 ∪ R2 )| ≤ λ R2 X Note: Analogous result holds for a unique minimal Rmin . I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 9/33

Figure 1: A graph G with 3 edge-disjoint (X , Y )-paths and the (X , Y )cuts Rmin and Rmax . I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 10/33

Important cuts Deﬁnition: δ(R) is the set of edges with exactly one endpoint in R. Deﬁnition: A set S of edges is an (X , Y )-cut 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 )-cut S can be expressed as S = δ(R) for some X ⊆ R and R ∩ Y = ∅. δ(R) Y X R I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 11/33

Important cuts Deﬁnition: An (X , Y )-cut δ(R) is important if there is no (X , Y )-cut δ(R ′ ) with R ⊂ R ′ and |δ(R ′ )| ≤ |δ(R)|. Note: Can be checked in polynomial time if a cut is important. δ(R) Y X R I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 11/33

Important cuts Deﬁnition: An (X , Y )-cut δ(R) is important if there is no (X , Y )-cut δ(R ′ ) with R ⊂ R ′ and |δ(R ′ )| ≤ |δ(R)|. Note: Can be checked in polynomial time if a cut is important. δ(R) Y X δ(R ′ ) R R′ I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 11/33

Important cuts Deﬁnition: An (X , Y )-cut δ(R) is important if there is no (X , Y )-cut δ(R ′ ) with R ⊂ R ′ and |δ(R ′ )| ≤ |δ(R)|. Note: Can be checked in polynomial time if a cut is important. δ(R) Y X R I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 11/33

Some Properties Let G be an undirected graph and X , Y ⊆ V (G ) two disjoint sets of vertices. Let S be an (X , Y )-cut and let R be the set of vertices reachable from X in G S. Then there is an important (X , Y )-cut S ′ = ∆(R ′ ) (possibly, S ′ = S) such that |S ′ | ≤ |S| and R ⊆ R ′ . Proof on board. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 12/33

Some Properties Let G be a graph, X , Y ⊆ V (G ) be two disjoint sets of vertices, and S = ∆(R) be an important (X , Y ) cut. 1. For every e ∈ S, the set S {e} is an important (X , Y )-cut in G e. 2. If S is an (X ′ , Y )-cut for some X ′ ⊃ X , then S is an important (X ′ , Y )-cut. Proof on board. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 13/33

Important cuts The number of important cuts can be exponentially large. Example: Y 1 k/2 2 X This graph has exactly 2k/2 important (X , Y )-cuts of size at most k. Theorem: There are at most 4k important (X , Y )-cuts of size at most k. (Proof is implicit in [Chen, Liu, Lu 2007], worse bound in [M. 2004].) I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 14/33

Important cuts Theorem: There are at most 4k important (X , Y )-cuts of size at most k. Proof: Let λ be the minimum (X , Y )-cut size and let δ(Rmax ) be the unique important cut of size λ such that Rmax is maximal. First we show that Rmax ⊆ R for every important cut δ(R). I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 15/33

Important cuts Theorem: There are at most 4k important (X , Y )-cuts of size at most k. Proof: Let λ be the minimum (X , Y )-cut size and let δ(Rmax ) be the unique important cut of size λ such that Rmax is maximal. First we show that Rmax ⊆ R for every important cut δ(R). By the submodularity of δ: |δ(Rmax )| + |δ(R)| ≥ |δ(Rmax ∩ R)| + |δ(Rmax ∪ R)| λ ≥λ ⇓ |δ(Rmax ∪ R)| ≤ |δ(R)| ⇓ If R = Rmax ∪ R, then δ(R) is not important. Thus the important (X , Y )- and (Rmax , Y )-cuts are the same. ⇒ We can assume X = Rmax . I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 15/33

Important cuts Theorem: There are at most 4k important (X , Y )-cuts of size at most k. Search tree algorithm for enumerating all these separators: An (arbitrary) edge uv leaving X = Rmax is either in the cut or not. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 16/33

Important cuts Theorem: There are at most 4k important (X , Y )-cuts of size at most k. Search tree algorithm for enumerating all these separators: An (arbitrary) edge uv leaving X = Rmax is either in the cut or not. Branch 1: If uv ∈ S, then S uv is an important (X , Y )-cut of size at most k − 1 in G uv . dsfsdfds Branch 2: If uv ∈ S, then S is an important (X ∪ v , Y )-cut of size at most k in G . X = Rmax u v Y dsfsdfds I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 16/33

Important cuts Theorem: There are at most 4k important (X , Y )-cuts of size at most k. Search tree algorithm for enumerating all these separators: An (arbitrary) edge uv leaving X = Rmax is either in the cut or not. Branch 1: If uv ∈ S, then S uv is an important (X , Y )-cut of size at most k − 1 in G uv . ⇒ k decreases by one, λ decreases by at most 1. Branch 2: If uv ∈ S, then S is an important (X ∪ v , Y )-cut of size at most k in G . X = Rmax u v Y ⇒ k remains the same, λ increases by 1. The measure 2k − λ decreases in each step. ⇒ Height of the search tree ≤ 2k ⇒ ≤ 22k important cuts of size ≤ k. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 16/33

Important cuts Example: The bound 4k is essentially tight. X Y I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 17/33

Important cuts Example: The bound 4k is essentially tight. X Y Any subtree with k leaves gives an important (X , Y )-cut of size k. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 17/33

Important cuts Example: The bound 4k is essentially tight. X Y Any subtree with k leaves gives an important (X , Y )-cut of size k. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 17/33

Important cuts Example: The bound 4k is essentially tight. X Y Any subtree with k leaves gives an important (X , Y )-cut of size k. The number of subtrees with k leaves is the Catalan number Ck−1 = 1 2k − 2 k k −1 ≥ 4k /poly(k). I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 17/33

Simple application Lemma: At most k · 4k edges incident to t can be part of an inclusionwise minimal s − t cut of size at most k. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 18/33

Simple application Lemma: At most k · 4k 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. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 18/33

Simple application Lemma: At most k · 4k 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)-cut δ(R ′ ) with R ⊆ R ′ and |δ(R ′ )| ≤ k. Clearly, vt ∈ δ(R ′ ): v ∈ R, hence v ∈ R ′ . I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 18/33

Anti isolation Let s, t1 , ... , tn be vertices and S1 , ... , Sn be sets of at most k edges such that Si separates ti from s, but Si does not separate tj from s for any j = i. It is possible that n is “large” even if k is “small.” t1 t2 t4 t3 t5 t6 s I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

Anti isolation Let s, t1 , ... , tn be vertices and S1 , ... , Sn be sets of at most k edges such that Si separates ti from s, but Si does not separate tj from s for any j = i. It is possible that n is “large” even if k is “small.” t1 t2 t3 t4 t5 t6 S1 s I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

Anti isolation Let s, t1 , ... , tn be vertices and S1 , ... , Sn be sets of at most k edges such that Si separates ti from s, but Si does not separate tj from s for any j = i. It is possible that n is “large” even if k is “small.” t1 t2 t3 t4 t5 t6 S2 s I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

Anti isolation Let s, t1 , ... , tn be vertices and S1 , ... , Sn be sets of at most k edges such that Si separates ti from s, but Si does not separate tj from s for any j = i. It is possible that n is “large” even if k is “small.” t1 t2 t4 t3 t5 t6 S3 s I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

Anti isolation Let s, t1 , ... , tn be vertices and S1 , ... , Sn be sets of at most k edges such that Si separates ti from s, but Si does not separate tj from s for any j = i. It is possible that n is “large” even if k is “small.” t1 t2 t4 t3 t5 t6 S1 s Is the opposite possible, i.e., Si separates every tj except ti ? I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

Anti isolation Let s, t1 , ... , tn be vertices and S1 , ... , Sn be sets of at most k edges such that Si separates ti from s, but Si does not separate tj from s for any j = i. It is possible that n is “large” even if k is “small.” t1 t2 t4 t3 t5 t6 S2 s Is the opposite possible, i.e., Si separates every tj except ti ? I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

Anti isolation Let s, t1 , ... , tn be vertices and S1 , ... , Sn be sets of at most k edges such that Si separates ti from s, but Si does not separate tj from s for any j = i. It is possible that n is “large” even if k is “small.” t1 t2 t4 t3 t5 t6 S3 s Is the opposite possible, i.e., Si separates every tj except ti ? I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

Anti isolation Let s, t1 , ... , tn be vertices and S1 , ... , Sn be sets of at most k edges such that Si separates ti from s, but Si does not separate tj from s for any j = i. It is possible that n is “large” even if k is “small.” t1 t2 t4 t3 t5 t6 S3 s Is the opposite possible, i.e., Si separates every tj except ti ? Lemma: If Si separates tj from s if and only j = i and every Si has size at most k, then n ≤ (k + 1) · 4k+1 . I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

Anti isolation t t1 t2 t3 t4 t5 t6 S3 s Is the opposite possible, i.e., Si separates every tj except ti ? Lemma: If Si separates tj from s if and only j = i and every Si has size at most k, then n ≤ (k + 1) · 4k+1 . Proof: Add a new vertex t. Every edge tti is part of an (inclusionwise minimal) (s, t)-separator of size at most k + 1. Use the previous lemma. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

Anti isolation t t1 t2 t3 t4 t5 t6 S2 s Is the opposite possible, i.e., Si separates every tj except ti ? Lemma: If Si separates tj from s if and only j = i and every Si has size at most k, then n ≤ (k + 1) · 4k+1 . Proof: Add a new vertex t. Every edge tti is part of an (inclusionwise minimal) (s, t)-separator of size at most k + 1. Use the previous lemma. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

Anti isolation t t1 t2 t3 t4 t5 t6 S1 s Is the opposite possible, i.e., Si separates every tj except ti ? Lemma: If Si separates tj from s if and only j = i and every Si has size at most k, then n ≤ (k + 1) · 4k+1 . Proof: Add a new vertex t. Every edge tti is part of an (inclusionwise minimal) (s, t)-separator of size at most k + 1. Use the previous lemma. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 19/33

M ULTIWAY C UT Deﬁnition: A multiway cut of a set of terminals T is a set S of edges such that each component of G S contains at most one vertex of T . t1 M ULTIWAY C UT Input: Graph G , set T of vertices, integer k Find: t2 t3 t5 A multiway cut S of at most k edges. t4 t4 Polynomial for |T | = 2, but NP-hard for any ﬁxed |T | ≥ 3 [Dalhaus et al. 1994]. Trivial to solve in polynomial time for ﬁxed k (in time nO(k) ). Theorem: M ULTIWAY CUT can be solved in time 4k · nO(1) , i.e., it is ﬁxed-parameter tractable (FPT) parameterized by the size k of the solution. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 20/33

M ULTIWAY C UT Intuition: Consider a t ∈ T . A subset of the solution S is a (t, T t)-cut. t I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 21/33

M ULTIWAY C UT Intuition: Consider a t ∈ T . A subset of the solution S is a (t, T t)-cut. t There are many such cuts. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 21/33

M ULTIWAY C UT Intuition: Consider a t ∈ T . A subset of the solution S is a (t, T t)-cut. t There are many such cuts. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 21/33

M ULTIWAY C UT Intuition: Consider a t ∈ T . A subset of the solution S is a (t, T t)-cut. t There are many such cuts. But a cut farther from t and closer to T t seems to be more useful. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 21/33

M ULTIWAY C UT and important separators Pushing Lemma: Let t ∈ T . The M ULTIWAY C UT problem has a solution S that contains an important (t, T t)-cut. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 22/33

M ULTIWAY C UT and important separators Pushing Lemma: Let t ∈ T . The M ULTIWAY C UT problem has a solution S that contains an important (t, T t)-cut. Proof: Let R be the vertices reachable from t in G S for a solution S. t R I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 22/33

M ULTIWAY C UT and important separators Pushing Lemma: Let t ∈ T . The M ULTIWAY C UT problem has a solution S that contains an important (t, T t)-cut. Proof: Let R be the vertices reachable from t in G S for a solution S. t R R′ If δ(R) is not important, then there is an important cut δ(R ′ ) with R ⊂ R ′ and |δ(R ′ )| ≤ |δ(R)|. Replace S with S ′ := (S δ(R)) ∪ δ(R ′ ) ⇒ |S ′ | ≤ |S| I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 22/33

M ULTIWAY C UT and important separators Pushing Lemma: Let t ∈ T . The M ULTIWAY C UT problem has a solution S that contains an important (t, T t)-cut. Proof: Let R be the vertices reachable from t in G S for a solution S. t u R v R′ If δ(R) is not important, then there is an important cut δ(R ′ ) with R ⊂ R ′ and |δ(R ′ )| ≤ |δ(R)|. Replace S with S ′ := (S δ(R)) ∪ δ(R ′ ) ⇒ |S ′ | ≤ |S| S ′ is a multiway cut: (1) There is no t-u path in G S ′ and (2) a u-v path in G S ′ implies a t-u path, a contradiction. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 22/33

M ULTIWAY C UT and important separators Pushing Lemma: Let t ∈ T . The M ULTIWAY C UT problem has a solution S that contains an important (t, T t)-cut. Proof: Let R be the vertices reachable from t in G S for a solution S. t u R v R′ If δ(R) is not important, then there is an important cut δ(R ′ ) with R ⊂ R ′ and |δ(R ′ )| ≤ |δ(R)|. Replace S with S ′ := (S δ(R)) ∪ δ(R ′ ) ⇒ |S ′ | ≤ |S| S ′ is a multiway cut: (1) There is no t-u path in G S ′ and (2) a u-v path in G S ′ implies a t-u path, a contradiction. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 22/33

Algorithm for M ULTIWAY C UT 1. If every vertex of T is in a different component, then we are done. 2. Let t ∈ T be a vertex that is not separated from every T t. 3. Branch on a choice of an important (t, T t) cut S of size at most k. 4. Set G := G S and k := k − |S|. 5. Go to step 1. We branch into at most 4k directions at most k times. (Better analysis gives 4k bound on the size of the search tree.) I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 23/33

M ULTICUT M ULTICUT Input: Graph G , pairs (s1 , t1 ), ... , (sℓ , tℓ ), integer k Find: A set S of edges such that G S has no si -ti path for any i. Theorem: M ULTICUT can be solved in time f (k, ℓ) · nO(1) (FPT parameterized by combined parameters k and ℓ). I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 24/33

M ULTICUT M ULTICUT Input: Graph G , pairs (s1 , t1 ), ... , (sℓ , tℓ ), integer k Find: A set S of edges such that G S has no si -ti path for any i. Theorem: M ULTICUT can be solved in time f (k, ℓ) · nO(1) (FPT parameterized by combined parameters k and ℓ). Proof: The solution partitions {s1 , t1 , ... , sℓ , tℓ } into components. Guess this partition, contract the vertices in a class, and solve M ULTIWAY C UT. Theorem: [Bousquet, Daligault, Thomassé 2011] [M., Razgon 2011] M ULTICUT is FPT parameterized by the size k of the solution. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 24/33

Directed graphs Deﬁnition: δ(R) is the set of edges leaving R. Observation: Every inclusionwise-minimal directed (X , Y )-cut S can be expressed as S = δ(R) for some X ⊆ R and R ∩ Y = ∅. Deﬁnition: An (X , Y )-cut δ(R) is important if there is no (X , Y )-cut δ(R ′ ) with R ⊂ R ′ and |δ(R ′ )| ≤ |δ(R)|. δ(R) X Y R I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 25/33

Directed graphs Deﬁnition: δ(R) is the set of edges leaving R. Observation: Every inclusionwise-minimal directed (X , Y )-cut S can be expressed as S = δ(R) for some X ⊆ R and R ∩ Y = ∅. Deﬁnition: An (X , Y )-cut δ(R) is important if there is no (X , Y )-cut δ(R ′ ) with R ⊂ R ′ and |δ(R ′ )| ≤ |δ(R)|. δ(R) X δ(R ′ ) Y R R′ I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 25/33

Directed graphs Deﬁnition: δ(R) is the set of edges leaving R. Observation: Every inclusionwise-minimal directed (X , Y )-cut S can be expressed as S = δ(R) for some X ⊆ R and R ∩ Y = ∅. Deﬁnition: An (X , Y )-cut δ(R) is important if there is no (X , Y )-cut δ(R ′ ) with R ⊂ R ′ and |δ(R ′ )| ≤ |δ(R)|. The proof for the undirected case goes through for the directed case: Theorem: There are at most 4k important directed (X , Y )-cuts of size at most k. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 25/33

D IRECTED M ULTIWAY C UT The undirected approach does not work: the pushing lemma is not true. Pushing Lemma: [for undirected graphs] Let t ∈ T . The M ULTIWAY C UT problem has a solution S that contains an important (t, T t)-cut. Directed counterexample: a s t b Unique solution with k = 1 edges, but it is not an important cut (boundary of {s, a}, but the boundary of {s, a, b} is of the same size). I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 26/33

D IRECTED M ULTIWAY C UT The undirected approach does not work: the pushing lemma is not true. Pushing Lemma: [for undirected graphs] Let t ∈ T . The M ULTIWAY C UT problem has a solution S that contains an important (t, T t)-cut. Directed counterexample: a s t b Unique solution with k = 1 edges, but it is not an important cut (boundary of {s, a}, but the boundary of {s, a, b} is of the same size). I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 26/33

D IRECTED M ULTIWAY C UT The undirected approach does not work: the pushing lemma is not true. Pushing Lemma: [for undirected graphs] Let t ∈ T . The M ULTIWAY C UT problem has a solution S that contains an important (t, T t)-cut. Directed counterexample: a s t b Unique solution with k = 1 edges, but it is not an important cut (boundary of {s, a}, but the boundary of {s, a, b} is of the same size). I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 26/33

D IRECTED M ULTIWAY C UT The undirected approach does not work: the pushing lemma is not true. Pushing Lemma: [for undirected graphs] Let t ∈ T . The M ULTIWAY C UT problem has a solution S that contains an important (t, T t)-cut. Problem in the undirected proof: t u v R R′ Replacing R by R ′ cannot create a t → u path, but can create a u → t path. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 26/33

D IRECTED M ULTIWAY C UT The undirected approach does not work: the pushing lemma is not true. Pushing Lemma: [for undirected graphs] Let t ∈ T . The M ULTIWAY C UT problem has a solution S that contains an important (t, T t)-cut. Problem in the undirected proof: t u v R R′ Replacing R by R ′ cannot create a t → u path, but can create a u → t path. Theorem: [Chitnis, Hajiaghayi, M. 2011] D IRECTED M ULTIWAY C UT is FPT parameterized by the size k of the solution. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 26/33

D IRECTED M ULTICUT D IRECTED M ULTICUT Input: Graph G , pairs (s1 , t1 ), ... , (sℓ , tℓ ), integer k Find: A set S of edges such that G S has no si → ti path for any i. Theorem: [M. and Razgon 2011] D IRECTED M ULTICUT is W[1]-hard parameterized by k. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 27/33

D IRECTED M ULTICUT D IRECTED M ULTICUT Input: Graph G , pairs (s1 , t1 ), ... , (sℓ , tℓ ), integer k Find: A set S of edges such that G S has no si → ti path for any i. Theorem: [M. and Razgon 2011] D IRECTED M ULTICUT is W[1]-hard parameterized by k. But the case ℓ = 2 can be reduced to D IRECTED M ULTIWAY C UT: s1 t1 t2 s2 I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 27/33

D IRECTED M ULTICUT D IRECTED M ULTICUT Input: Graph G , pairs (s1 , t1 ), ... , (sℓ , tℓ ), integer k Find: A set S of edges such that G S has no si → ti path for any i. Theorem: [M. and Razgon 2011] D IRECTED M ULTICUT is W[1]-hard parameterized by k. But the case ℓ = 2 can be reduced to D IRECTED M ULTIWAY C UT: x s1 t1 t2 y s2 I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 27/33

D IRECTED M ULTICUT D IRECTED M ULTICUT Input: Graph G , pairs (s1 , t1 ), ... , (sℓ , tℓ ), integer k Find: A set S of edges such that G S has no si → ti path for any i. Theorem: [M. and Razgon 2011] D IRECTED M ULTICUT is W[1]-hard parameterized by k. But the case ℓ = 2 can be reduced to D IRECTED M ULTIWAY C UT: x s1 t1 t2 y s2 I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 27/33

D IRECTED M ULTICUT D IRECTED M ULTICUT Input: Graph G , pairs (s1 , t1 ), ... , (sℓ , tℓ ), integer k Find: A set S of edges such that G S has no si → ti path for any i. Theorem: [M. and Razgon 2011] D IRECTED M ULTICUT is W[1]-hard parameterized by k. Corollary: D IRECTED M ULTICUT with ℓ = 2 is FPT parameterized by the size k of the solution. ? Open: Is D IRECTED M ULTICUT with ℓ = 3 FPT? Open: Is there an f (k, ℓ) · nO(1) algorithm for D IRECTED M ULTICUT? I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 27/33

S KEW M ULTICUT S KEW M ULTICUT Input: Graph G , pairs (s1 , t1 ), ... , (sℓ , tℓ ), integer k Find: A set S of k directed edges such that G S contains no si → tj path for any i ≤ j. s1 t1 s2 t2 s3 t3 s4 t4 I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 28/33

S KEW M ULTICUT S KEW M ULTICUT Input: Graph G , pairs (s1 , t1 ), ... , (sℓ , tℓ ), integer k Find: A set S of k directed edges such that G S contains no si → tj path for any i ≤ j. s1 t1 s2 t2 s3 t3 s4 t4 Pushing Lemma: S KEW M ULTCUT problem has a solution S that contains an important (s1 , {t1 , ... , tℓ })-cut. Theorem: [Chen, Liu, Lu, O’Sullivan, Razgon 2008] S KEW M ULTICUT can be solved in time 4k · nO(1) . I ,I P – p. 28/33 MPORTANT CUTS MPORTANT SEPARATORS AND ARAMETERIZED ALGORITHMS

D IRECTED F EEDBACK V ERTEX S ET D IRECTED F EEDBACK V ERTEX /E DGE S ET Input: Directed graph G , integer k Find: A set S of k vertices/edges such that G S is acyclic. Note: Edge and vertex versions are equivalent, we will consider the edge version here. Theorem: [Chen, Liu, Lu, O’Sullivan, Razgon 2008] D IRECTED F EEDBACK E DGE S ET is FPT parameterized by the size k of the solution. Solution uses the technique of Smith, Vetta 2004]. iterative compress ion introduced by [Reed, I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 29/33

The compression problem D IRECTED F EEDBACK E DGE S ET C OMPRESSION Input: Directed graph G , integer k, a set S ′ of k + 1 edges such that G S ′ is acyclic Find: A set S of k edges such that G S is acyclic. Easier than the original problem, as the extra input S ′ gives us useful structural information about G . Lemma: The compression problem is FPT parameterized by k. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 30/33

The compression problem Lemma: The compression problem is FPT parameterized by k. −→ − −− −→ −−− Proof: Let S ′ = {t1 s1 , ... , tk+1 sk+1 }. t 4 s4 t 3 s3 t 2 s2 t 1 s1 By guessing and removing S ∩ S ′ , we can assume that S and S ′ are disjoint [2k+1 possibilities]. By guessing the order of {s1 , ... , sk+1 } in the acyclic ordering of G S, we can assume that sk+1 < sk < · · · < s1 in G S [(k + 1)! possibilities]. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 30/33

The compression problem Lemma: The compression problem is FPT parameterized by k. −→ − −− −→ −−− Proof: Let S ′ = {t1 s1 , ... , tk+1 sk+1 }. t 4 s4 t 3 s3 t 2 s2 t 1 s1 Claim: Suppose that S ′ ∩ S = ∅. G S is acyclic and has an ordering with sk+1 < sk < · · · < s1 S covers every si → tj path for every i ≤ j I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 30/33

The compression problem Lemma: The compression problem is FPT parameterized by k. −→ − −− −→ −−− Proof: Let S ′ = {t1 s1 , ... , tk+1 sk+1 }. t 4 s4 t 3 s3 t 2 s2 t 1 s1 Claim: Suppose that S ′ ∩ S = ∅. G S is acyclic and has an ordering with sk+1 < sk < · · · < s1 S covers every si → tj path for every i ≤ j I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 30/33

The compression problem Lemma: The compression problem is FPT parameterized by k. −→ − −− −→ −−− Proof: Let S ′ = {t1 s1 , ... , tk+1 sk+1 }. t 4 s4 t 3 s3 t 2 s2 t 1 s1 Claim: Suppose that S ′ ∩ S = ∅. G S is acyclic and has an ordering with sk+1 < sk < · · · < s1 S covers every si → tj path for every i ≤ j ⇒ We can solve the compression problem by 2k+1 · (k + 1)! applications of S KEW M ULTICUT. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 30/33

Iterative compression We have given a f (k)nO(1) algorithm for the following problem: D IRECTED F EEDBACK E DGE S ET C OMPRESSION Input: Directed graph G , integer k, a set S ′ of k + 1 edges such that G S ′ is acyclic Find: A set S of k edges such that G S is acyclic. Nice, but how do we get a solution S ′ of size k + 1? I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 31/33

Iterative compression We have given a f (k)nO(1) algorithm for the following problem: D IRECTED F EEDBACK E DGE S ET C OMPRESSION Input: Directed graph G , integer k, a set S ′ of k + 1 edges such that G S ′ is acyclic Find: A set S of k edges such that G S is acyclic. Nice, but how do we get a solution S ′ of size k + 1? We get it for free! it Useful trick: erative compression (introduced by [Reed, Smith, Vetta 2004] for B IPARTITE D ELETION). I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 31/33

Iterative compression Let e1 , ... , em be the edges of G and let Gi be the subgraph containing only the ﬁrst i edges (and all vertices). For every i = 1, ... , m, we ﬁnd a set Si of k edges such that Gi Si is acyclic.

Iterative compression Let e1 , ... , em be the edges of G and let Gi be the subgraph containing only the ﬁrst i edges (and all vertices). For every i = 1, ... , m, we ﬁnd a set Si of k edges such that Gi Si is acyclic. For i = k, we have the trivial solution Si = {e1 , ... , ek }. Suppose we have a solution Si for Gi . Then Si ∪ {ei+1 } is a solution of size k + 1 in the graph Gi+1 Use the compression algorithm for Gi+1 with the solution Si ∪ {ei+1 }. If there is no solution of size k for Gi+1 , then we can stop. Otherwise the compression algorithm gives a solution Si+1 of size k for Gi+1 . We call the compression algorithm m times, everything else is polynomial. ⇒ D IRECTED F EEDBACK E DGE S ET is FPT. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 32/33

Conclusions A simple (but essentially tight) bound on the number of important cuts. Algorithmic results: FPT algorithms for M ULTIWAY C UT in undirected graphs, S KEW M ULTICUT in directed graphs, and D IRECTED F EEDBACK V ERTEX /E DGE S ET. I MPORTANT CUTS , I MPORTANT SEPARATORS AND PARAMETERIZED ALGORITHMS – p. 33/33

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