Information about Treewidth and Applications

Introduction and deﬁnition DONE Part I: Algorithms for bounded treewidth graphs. DONE Part II: Graph-theoretic properties of treewidth. Part III: Applications for general graphs. Part IV: Algorithm for ﬁnding treewidth Part V: Irrelevant Vertices – Planar Vertex Deletion.

Applications Algorithms for graphs of bounded treewidth ﬁnd many applications, for example in Graph Minors Exact Algorithms Approximation Schemes Parameterized Algorithms Kernelization Databases CSP’s

Application I: Parameterized Algorithms Extended Monadic Second Order Logic (EMSO) A logical language on graphs consisting of the following: Logical connectives ∧, ∨, →, ¬, =, = quantiﬁers ∀, ∃ over vertex/edge variables predicate adj(u, v ): vertices u and v are adjacent predicate inc(e, v ): edge e is incident to vertex v ∈, ⊆ for vertex/edge sets Example: The formula ∃C ⊆ V ∀v ∈ C ∃u1 , u2 ∈ C (u1 = u2 ∧ adj(u1 , v ) ∧ adj(u2 , v )) is true . . .if graph G (V , E ) has a cycle.

Application I: Parameterized Algorithms Extended Monadic Second Order Logic (EMSO) A logical language on graphs consisting of the following: Logical connectives ∧, ∨, →, ¬, =, = quantiﬁers ∀, ∃ over vertex/edge variables predicate adj(u, v ): vertices u and v are adjacent predicate inc(e, v ): edge e is incident to vertex v ∈, ⊆ for vertex/edge sets Example: The formula ∃C ⊆ V ∀v ∈ C ∃u1 , u2 ∈ C (u1 = u2 ∧ adj(u1 , v ) ∧ adj(u2 , v )) is true . . .if graph G (V , E ) has a cycle.

Application I: Parameterized Algorithms Extended Monadic Second Order Logic (EMSO) A logical language on graphs consisting of the following: Logical connectives ∧, ∨, →, ¬, =, = quantiﬁers ∀, ∃ over vertex/edge variables predicate adj(u, v ): vertices u and v are adjacent predicate inc(e, v ): edge e is incident to vertex v ∈, ⊆ for vertex/edge sets Example: The formula ∃C ⊆ V ∀v ∈ C ∃u1 , u2 ∈ C (u1 = u2 ∧ adj(u1 , v ) ∧ adj(u2 , v )) is true . . .if graph G (V , E ) has a cycle.

Courcelle’s Theorem Courcelle’s Theorem: If a graph property can be expressed in EMSO with formula ϕ of size |ϕ|, then for every ﬁxed w ≥ 1, there is an algorithm running in time f (|ϕ|, w ) · n for testing this property on graphs having treewidth at most w . Independent Set, Dominating Set, q Coloring, Max Cut, Odd Cycle Transversal, Hamiltonian Cycle, Partition into Triangles, Feedback Vertex Set, Vertex Disjoint Cycle Packing and million other problems are FPT parameterized by the treewidth.

Courcelle’s Theorem Courcelle’s Theorem: If a graph property can be expressed in EMSO with formula ϕ of size |ϕ|, then for every ﬁxed w ≥ 1, there is an algorithm running in time f (|ϕ|, w ) · n for testing this property on graphs having treewidth at most w . Independent Set, Dominating Set, q Coloring, Max Cut, Odd Cycle Transversal, Hamiltonian Cycle, Partition into Triangles, Feedback Vertex Set, Vertex Disjoint Cycle Packing and million other problems are FPT parameterized by the treewidth.

Properties of treewidth Fact: treewidth ≤ 2 if and only if graph is subgraph of a series-parallel graph

Properties of treewidth Fact: treewidth ≤ 2 if and only if graph is subgraph of a series-parallel graph Fact: For every k ≥ 2, the treewidth of the k × k grid is exactly k.

Properties of treewidth Fact: treewidth ≤ 2 if and only if graph is subgraph of a series-parallel graph Fact: For every k ≥ 2, the treewidth of the k × k grid is exactly k. Fact: Treewidth does not increase if we delete edges, delete vertices, or contract edges. =⇒ If F is a minor of G , then the treewidth of F is at most the treewidth of G .

Properties of treewidth Fact: treewidth ≤ 2 if and only if graph is subgraph of a series-parallel graph Fact: For every k ≥ 2, the treewidth of the k × k grid is exactly k. Fact: Treewidth does not increase if we delete edges, delete vertices, or contract edges. =⇒ If F is a minor of G , then the treewidth of F is at most the treewidth of G . The treewidth of the k-clique is k − 1.

Obstruction to Treewidth Figure : Example of a 6 × 6-grid 6 and a triangulated grid Γ4 . Another, extremely useful obstructions to small treewidth, are grid-minors. Let t be a positive integer. The t × t-grid t is a graph with vertex set {(x, y ) | x, y ∈ {1, 2, . . . , t}}. Thus t has exactly t 2 vertices. Two diﬀerent vertices (x, y ) and (x , y ) are adjacent if and only if |x − x | + |y − y | ≤ 1. The border of t is the set of vertices with coordinates (1, y ), (t, y ), (t, 1), and (x, t), where x, y ∈ {1, 2, . . . , t}

As we already have seen, if a graph contains large grid as a minor, its treewidth is also large.

As we already have seen, if a graph contains large grid as a minor, its treewidth is also large. What is much more surprising, is that the converse is also true, every graph of large treewidth contains a large grid as a minor.

Excluded Grid Theorem Fact: [Excluded Grid Theorem] If the treewidth of G is at least 2 k 4k (k+2) , then G has a k × k grid minor. [Robertson and Seymour ]

Excluded Grid Theorem Fact: [Excluded Grid Theorem] If the treewidth of G is at least 2 k 4k (k+2) , then G has a k × k grid minor. [Robertson and Seymour ] It was open for many years whether a polynomial relationship could be established between the treewidth of a graph G and the size of its largest grid minor.

Excluded Grid Theorem Fact: [Excluded Grid Theorem] If the treewidth of G is at least 2 k 4k (k+2) , then G has a k × k grid minor. [Robertson and Seymour ] It was open for many years whether a polynomial relationship could be established between the treewidth of a graph G and the size of its largest grid minor. Theorem (Excluded Grid Theorem, Chekuri and Chuzhoy) Let t ≥ 0 be an integer. There exists a universal constant c, such that every graph of treewidth at least c · t 99 contains t as a minor.

Excluded Grid Theorem Fact: [Excluded Grid Theorem] If the treewidth of G is at least 2 k 4k (k+2) , then G has a k × k grid minor. [Robertson and Seymour ]

Excluded Grid Theorem Fact: [Excluded Grid Theorem] If the treewidth of G is at least 2 k 4k (k+2) , then G has a k × k grid minor. [Robertson and Seymour ] It was open for many years whether a polynomial relationship could be established between the treewidth of a graph G and the size of its largest grid minor.

Excluded Grid Theorem Fact: [Excluded Grid Theorem] If the treewidth of G is at least 2 k 4k (k+2) , then G has a k × k grid minor. [Robertson and Seymour ] It was open for many years whether a polynomial relationship could be established between the treewidth of a graph G and the size of its largest grid minor. Theorem (Excluded Grid Theorem, Chekuri and Chuzhoy) Let t ≥ 0 be an integer. There exists a universal constant c, such that every graph of treewidth at least c · t 99 contains t as a minor.

Excluded Grid Theorem A : Planar Graph Much better relations. We have two theorems: Theorem (Planar Excluded Grid Theorem) Let t ≥ 0 be an integer. Every planar graph G of treewidth at least 9 t, contains t as a minor. Furthermore, there exists a 2 polynomial-time algorithm that for a given planar graph G either outputs a tree decomposition of G of width 9 t or constructs a 2 minor model of t in G .

Excluded Grid Theorem : Planar Graph One more Excluded Grid Theorem, this time not for minors but just for edge contractions. Figure : Example of a 6 × 6-grid 6 and a triangulated grid Γ4 .

Excluded Grid Theorem : Planar Graph One more Excluded Grid Theorem, this time not for minors but just for edge contractions. Figure : Example of a 6 × 6-grid 6 and a triangulated grid Γ4 . For an integer t > 0 the graph Γt is obtained from the grid t by adding for every 1 ≤ x, y ≤ t − 1, the edge (x, y ), (x + 1, y + 1), and making the vertex (t, t) adjacent to all vertices with x ∈ {1, t} and y ∈ {1, t}.

Excluded Grid Theorem : Planar Graph Figure : Example of a 6 × 6-grid 6 and a triangulated grid Γ4 . Theorem For any connected planar graph G and integer t ≥ 0, if tw(G ) ≥ 9(t + 1), then G contains Γt as a contraction. Furthermore there exists a polynomial-time algorithm that given G either outputs a tree decomposition of G of width 9(t + 1) or a set of edges whose contraction result in Γt .

Excluded Grid Theorem : Planar Graph One more Excluded Grid Theorem, this time not for minors but just for edge contractions. Theorem For any connected planar graph G and integer t ≥ 0, if tw(G ) ≥ 9(t + 1), then G contains Γt as a contraction. Furthermore there exists a polynomial-time algorithm that given G either outputs a tree decomposition of G of width 9(t + 1) or a set of edges whose contraction result in Γt .

Excluded Grid Theorem : Planar Graph One more Excluded Grid Theorem, this time not for minors but just for edge contractions. Theorem For any connected planar graph G and integer t ≥ 0, if tw(G ) ≥ 9(t + 1), then G contains Γt as a contraction. Furthermore there exists a polynomial-time algorithm that given G either outputs a tree decomposition of G of width 9(t + 1) or a set of edges whose contraction result in Γt . Can we have such theorems for general graphs?

Outerplanar graphs Deﬁnition: A planar graph is outerplanar if it has a planar embedding where every vertex is on the inﬁnite face.

Outerplanar graphs Deﬁnition: A planar graph is outerplanar if it has a planar embedding where every vertex is on the inﬁnite face. Fact: Every outerplanar graph has treewidth at most 2. =⇒ Every outerplanar graph is series-parallel.

k-outerplanar graphs Given a planar embedding, we can deﬁne layers by iteratively removing the vertices on the inﬁnite face. Deﬁnition: A planar graph is k-outerplanar if it has a planar embedding having at most k layers. 1 1 2 1 2 1 2 3 2 3 2 3 3 3 3 3 2 2 2 2 2 1 1 Fact: Every k-outerplanar graph has treewidth at most 3k + 1.

k-outerplanar graphs Given a planar embedding, we can deﬁne layers by iteratively removing the vertices on the inﬁnite face. Deﬁnition: A planar graph is k-outerplanar if it has a planar embedding having at most k layers. 1 1 2 1 2 1 2 3 2 3 2 3 3 3 3 3 2 2 2 2 2 1 1 Fact: Every k-outerplanar graph has treewidth at most 3k + 1.

k-outerplanar graphs Given a planar embedding, we can deﬁne layers by iteratively removing the vertices on the inﬁnite face. Deﬁnition: A planar graph is k-outerplanar if it has a planar embedding having at most k layers. 2 2 2 3 2 3 2 3 3 3 3 3 2 2 2 2 2 Fact: Every k-outerplanar graph has treewidth at most 3k + 1.

k-outerplanar graphs Given a planar embedding, we can deﬁne layers by iteratively removing the vertices on the inﬁnite face. Deﬁnition: A planar graph is k-outerplanar if it has a planar embedding having at most k layers. 2 2 2 3 2 3 2 3 3 3 3 3 2 2 2 2 2 Fact: Every k-outerplanar graph has treewidth at most 3k + 1.

k-outerplanar graphs Given a planar embedding, we can deﬁne layers by iteratively removing the vertices on the inﬁnite face. Deﬁnition: A planar graph is k-outerplanar if it has a planar embedding having at most k layers. 3 3 3 3 3 3 3 Fact: Every k-outerplanar graph has treewidth at most 3k + 1.

Shifting Techniques

Building Blocks of the Technique r For vertex v of a graph G and integer r ≥ 1, we denote by Gv the subgraph of G induced by vertices within distance r from v in G .

Building Blocks of the Technique r For vertex v of a graph G and integer r ≥ 1, we denote by Gv the subgraph of G induced by vertices within distance r from v in G . Lemma Let G be a planar graph, v ∈ V (G ) and r ≥ 1. Then r tw(Gv ) ≤ 18(r + 1). Proof. On board.

Building Blocks of the Technique r For vertex v of a graph G and integer r ≥ 1, we denote by Gv the subgraph of G induced by vertices within distance r from v in G . Lemma Let G be a planar graph, v ∈ V (G ) and r ≥ 1. Then r tw(Gv ) ≤ 18(r + 1). Proof. On board. 18(r + 1) in the above proof can be made 3r + 1. Lemma Let v be a vertex of a planar graph G and let Li , be the vertices of G at distance i, 0 ≤ i ≤ n, from v . Then for any i, j ≥ 0, the treewidth of the subgraph Gi,i+j induced by vertices in Li ∪ Li+1 ∪ · · · ∪ Li+ j does not exceed 3j + 1. Proof. On board.

Intuition The idea behind the shifting technique is as follows. Pick a vertex v of planar graph G and run breadth-ﬁrst search (BFS) from v . For any i, j ≥ 0, the treewidth of the subgraph Gi,i+j induced by vertices in levels i, i + 1, . . . , i + j of BFS does not exceed 3j + 1. Now for an appropriate choice of parameters, we can ﬁnd a“shift” of “windows”, i.e. a disjoint set of a small number of consecutive levels of BFS, “covering” the solution. Because every window is of small treewidth, we can employ the dynamic programing or the power of Courcelle’s theorem to solve the problem. We will see two examples.

Intuition The idea behind the shifting technique is as follows. Pick a vertex v of planar graph G and run breadth-ﬁrst search (BFS) from v . For any i, j ≥ 0, the treewidth of the subgraph Gi,i+j induced by vertices in levels i, i + 1, . . . , i + j of BFS does not exceed 3j + 1. Now for an appropriate choice of parameters, we can ﬁnd a“shift” of “windows”, i.e. a disjoint set of a small number of consecutive levels of BFS, “covering” the solution. Because every window is of small treewidth, we can employ the dynamic programing or the power of Courcelle’s theorem to solve the problem. We will see two examples.

Intuition The idea behind the shifting technique is as follows. Pick a vertex v of planar graph G and run breadth-ﬁrst search (BFS) from v . For any i, j ≥ 0, the treewidth of the subgraph Gi,i+j induced by vertices in levels i, i + 1, . . . , i + j of BFS does not exceed 3j + 1. Now for an appropriate choice of parameters, we can ﬁnd a“shift” of “windows”, i.e. a disjoint set of a small number of consecutive levels of BFS, “covering” the solution. Because every window is of small treewidth, we can employ the dynamic programing or the power of Courcelle’s theorem to solve the problem. We will see two examples.

Intuition The idea behind the shifting technique is as follows. Pick a vertex v of planar graph G and run breadth-ﬁrst search (BFS) from v . For any i, j ≥ 0, the treewidth of the subgraph Gi,i+j induced by vertices in levels i, i + 1, . . . , i + j of BFS does not exceed 3j + 1. Now for an appropriate choice of parameters, we can ﬁnd a“shift” of “windows”, i.e. a disjoint set of a small number of consecutive levels of BFS, “covering” the solution. Because every window is of small treewidth, we can employ the dynamic programing or the power of Courcelle’s theorem to solve the problem. We will see two examples.

Subgraph Isomorphism Subgraph Isomorphism: given graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. EMSO formula of size k O(1) for Subgraph Isomorphism exists. This now using Courcelle’s Theorem implies that we have f (k, w ) · n time algorithm for Subgraph Isomorphism on graphs of treewidth w .

Subgraph Isomorphism Subgraph Isomorphism: given graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. EMSO formula of size k O(1) for Subgraph Isomorphism exists. This now using Courcelle’s Theorem implies that we have f (k, w ) · n time algorithm for Subgraph Isomorphism on graphs of treewidth w .

Subgraph Isomorphism Subgraph Isomorphism: given graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. EMSO formula of size k O(1) for Subgraph Isomorphism exists. This now using Courcelle’s Theorem implies that we have f (k, w ) · n time algorithm for Subgraph Isomorphism on graphs of treewidth w .

Baker’s shifting strategy Subgraph Isomorphism for planar graphs: given planar graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. Layers of the planar graph: (as in the deﬁnition of k-outerplanar): For a ﬁxed 0 ≤ s < k + 1, delete every layer Li with i = s (mod k + 1) The resulting graph is k-outerplanar, hence it has treewidth at most 3k + 1. Using the f (k, w ) · n time algorithm for Subgraph Isomorphism, the problem can be solved in time f (k, 3k + 1) · n. We do this for every 0 ≤ s < k + 1: for at least one value of s, we do not delete any of the k vertices of the solution =⇒ we ﬁnd a copy of H in G if there is one. Subgraph Isomorphism for planar graphs is FPT parameterized by k := |V (H)|.

Baker’s shifting strategy Subgraph Isomorphism for planar graphs: given planar graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. Layers of the planar graph: (as in the deﬁnition of k-outerplanar): For a ﬁxed 0 ≤ s < k + 1, delete every layer Li with i = s (mod k + 1) The resulting graph is k-outerplanar, hence it has treewidth at most 3k + 1. Using the f (k, w ) · n time algorithm for Subgraph Isomorphism, the problem can be solved in time f (k, 3k + 1) · n. We do this for every 0 ≤ s < k + 1: for at least one value of s, we do not delete any of the k vertices of the solution =⇒ we ﬁnd a copy of H in G if there is one. Subgraph Isomorphism for planar graphs is FPT parameterized by k := |V (H)|.

Baker’s shifting strategy Subgraph Isomorphism for planar graphs: given planar graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. Layers of the planar graph: (as in the deﬁnition of k-outerplanar): For a ﬁxed 0 ≤ s < k + 1, delete every layer Li with i = s (mod k + 1) The resulting graph is k-outerplanar, hence it has treewidth at most 3k + 1. Using the f (k, w ) · n time algorithm for Subgraph Isomorphism, the problem can be solved in time f (k, 3k + 1) · n. We do this for every 0 ≤ s < k + 1: for at least one value of s, we do not delete any of the k vertices of the solution =⇒ we ﬁnd a copy of H in G if there is one. Subgraph Isomorphism for planar graphs is FPT parameterized by k := |V (H)|.

Baker’s shifting strategy Subgraph Isomorphism for planar graphs: given planar graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. Layers of the planar graph: (as in the deﬁnition of k-outerplanar): For a ﬁxed 0 ≤ s < k + 1, delete every layer Li with i = s (mod k + 1) The resulting graph is k-outerplanar, hence it has treewidth at most 3k + 1. Using the f (k, w ) · n time algorithm for Subgraph Isomorphism, the problem can be solved in time f (k, 3k + 1) · n. We do this for every 0 ≤ s < k + 1: for at least one value of s, we do not delete any of the k vertices of the solution =⇒ we ﬁnd a copy of H in G if there is one. Subgraph Isomorphism for planar graphs is FPT parameterized by k := |V (H)|.

Baker’s shifting strategy Subgraph Isomorphism for planar graphs: given planar graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. Layers of the planar graph: (as in the deﬁnition of k-outerplanar): For a ﬁxed 0 ≤ s < k + 1, delete every layer Li with i = s (mod k + 1) The resulting graph is k-outerplanar, hence it has treewidth at most 3k + 1. Using the f (k, w ) · n time algorithm for Subgraph Isomorphism, the problem can be solved in time f (k, 3k + 1) · n. We do this for every 0 ≤ s < k + 1: for at least one value of s, we do not delete any of the k vertices of the solution =⇒ we ﬁnd a copy of H in G if there is one. Subgraph Isomorphism for planar graphs is FPT parameterized by k := |V (H)|.

Baker’s shifting strategy Subgraph Isomorphism for planar graphs: given planar graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. Layers of the planar graph: (as in the deﬁnition of k-outerplanar): For a ﬁxed 0 ≤ s < k + 1, delete every layer Li with i = s (mod k + 1) The resulting graph is k-outerplanar, hence it has treewidth at most 3k + 1. Using the f (k, w ) · n time algorithm for Subgraph Isomorphism, the problem can be solved in time f (k, 3k + 1) · n. We do this for every 0 ≤ s < k + 1: for at least one value of s, we do not delete any of the k vertices of the solution =⇒ we ﬁnd a copy of H in G if there is one. Subgraph Isomorphism for planar graphs is FPT parameterized by k := |V (H)|.

Baker’s shifting strategy Subgraph Isomorphism for planar graphs: given planar graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. Layers of the planar graph: (as in the deﬁnition of k-outerplanar): For a ﬁxed 0 ≤ s < k + 1, delete every layer Li with i = s (mod k + 1) The resulting graph is k-outerplanar, hence it has treewidth at most 3k + 1. Using the f (k, w ) · n time algorithm for Subgraph Isomorphism, the problem can be solved in time f (k, 3k + 1) · n. We do this for every 0 ≤ s < k + 1: for at least one value of s, we do not delete any of the k vertices of the solution =⇒ we ﬁnd a copy of H in G if there is one. Subgraph Isomorphism for planar graphs is FPT parameterized by k := |V (H)|.

Baker’s shifting strategy Subgraph Isomorphism for planar graphs: given planar graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. Layers of the planar graph: (as in the deﬁnition of k-outerplanar): For a ﬁxed 0 ≤ s < k + 1, delete every layer Li with i = s (mod k + 1) The resulting graph is k-outerplanar, hence it has treewidth at most 3k + 1. Using the f (k, w ) · n time algorithm for Subgraph Isomorphism, the problem can be solved in time f (k, 3k + 1) · n. We do this for every 0 ≤ s < k + 1: for at least one value of s, we do not delete any of the k vertices of the solution =⇒ we ﬁnd a copy of H in G if there is one. Subgraph Isomorphism for planar graphs is FPT parameterized by k := |V (H)|.

Baker’s shifting strategy Subgraph Isomorphism for planar graphs: given planar graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. Layers of the planar graph: (as in the deﬁnition of k-outerplanar): For a ﬁxed 0 ≤ s < k + 1, delete every layer Li with i = s (mod k + 1) The resulting graph is k-outerplanar, hence it has treewidth at most 3k + 1. Using the f (k, w ) · n time algorithm for Subgraph Isomorphism, the problem can be solved in time f (k, 3k + 1) · n. We do this for every 0 ≤ s < k + 1: for at least one value of s, we do not delete any of the k vertices of the solution =⇒ we ﬁnd a copy of H in G if there is one. Subgraph Isomorphism for planar graphs is FPT parameterized by k := |V (H)|.

Baker’s shifting strategy Subgraph Isomorphism for planar graphs: given planar graphs H and G , ﬁnd a copy of H in G as subgraph. Parameter k := |V (H)|. Layers of the planar graph: (as in the deﬁnition of k-outerplanar): For a ﬁxed 0 ≤ s < k + 1, delete every layer Li with i = s (mod k + 1) The resulting graph is k-outerplanar, hence it has treewidth at most 3k + 1. Using the f (k, w ) · n time algorithm for Subgraph Isomorphism, the problem can be solved in time f (k, 3k + 1) · n. We do this for every 0 ≤ s < k + 1: for at least one value of s, we do not delete any of the k vertices of the solution =⇒ we ﬁnd a copy of H in G if there is one. Subgraph Isomorphism for planar graphs is FPT parameterized by k := |V (H)|.

Bisection For a given n-vertex graph G , weight function w : V (G ) → N and integer k, the task is to decide if there is a partition of V (G ) into sets V1 and V2 of weights w (V (G ))/2 and w (V (G )/2 and such that the number of edges between V1 and V2 is at most k. In other words, we are looking for a balanced partition (V1 , V2 ) with a (V1 , V2 )-cut at most k.

Bisection For a given n-vertex graph G , weight function w : V (G ) → N and integer k, the task is to decide if there is a partition of V (G ) into sets V1 and V2 of weights w (V (G ))/2 and w (V (G )/2 and such that the number of edges between V1 and V2 is at most k. In other words, we are looking for a balanced partition (V1 , V2 ) with a (V1 , V2 )-cut at most k. Lemma Bisection is solvable in time 2t · nO(1) on an n-vertex given together with its tree decomposition of width t. Theorem Bisection on planar graphs is solvable in time 2O(k) · nO(1) . Proof. On board.

Approximation schemes Deﬁnition: A polynomial-time approximation scheme (PTAS) for a problem P is an algorithm that takes an instance of P and a rational number ε > 0, always ﬁnds a (1 + ε)-approximate solution, the running time is polynomial in n for every ﬁxed ε > 0. 2 Typical running times: 21/ε · n, n1/ε , (n/ε)2 , n1/ε . Some classical problems that have a PTAS: Independent Set for planar graphs TSP in the Euclidean plane Steiner Tree in planar graphs Knapsack

Baker’s shifting strategy for EPTAS Fact: There is a 2O(1/ε) · n time PTAS for Independent Set for planar graphs. Let D := 1/ε. For a ﬁxed 0 ≤ s < D, delete every layer Li with i = s (mod D) The resulting graph is D-outerplanar, hence it has treewidth at most 3D + 1 = O(1/ε). Using the O(2w · n) time algorithm for Independent Set, the problem can be solved in time 2O(1/ε) · n. We do this for every 0 ≤ s < D: for at least one value of s, we delete at most 1/D = ε fraction of the solution =⇒ we get a (1 − ε)-approximate solution.

Bidimensionality

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