Algorithms for Graphs of Bounded Treewidth Made by Moshe Sebag CS - - PowerPoint PPT Presentation

Algorithms for Graphs of Bounded Treewidth

Made by Moshe Sebag CS department, Technion

The material for all parts of this lecture appears in Chapter 7 of the book "Parameterized Algorithms", by Cygan et al.

Table of Content

 Into: explanation of the term "treewidth of a graph“  Definitions:

• 1. Tree decomposition of a graph
• 2. Nice tree decomposition

 Dynamic Programming (DP) on graphs of bounded treewidth  Example of DP-based algorithm for Weighted Independent Set  Run-time analysis of the algorithm  Treewidth and Monadic second-order Logic  Courcelle's theorem

Treewidth of a Graph

 The treewidth of an undirected graph is a number associated with the graph.  Very roughly, treewidth captures how similar a graph is to a tree.  Treewidth is commonly used as a parameter in the parameterized complexity analysis of graph algorithms.  In this lecture, we focus on connections to the idea of dynamic programming on the structure of a graph.

Treewidth of a Graph

 The treewidth of an undirected graph is a number associated with the graph.  Very roughly, treewidth captures how similar a graph is to a tree.  Treewidth is commonly used as a parameter in the parameterized complexity analysis of graph algorithms.  In this lecture, we focus on connections to the idea of dynamic programming on the structure of a graph.

But how do you calculate the treewidth of a graph?

Tree Decomposition

 a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph.

Tree Decomposition

 Definition: a tree decomposition of a graph 𝐻 is a pair 𝓤 = (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ), where 𝑈 is a tree whose every node 𝑢 is assigned a vertex subset 𝑌𝑢 ⊆ 𝑊 (𝐻).  The following three conditions hold:

 (T1)ڂ𝑢∈𝑊(𝑈) 𝑌𝑢 = 𝑊(𝐻)  (T2) For every 𝑣𝑤 ∈ 𝐹(𝐻) , there exists a node 𝑢 of 𝑈 such that bag 𝑌𝑢 contains both 𝑣 and 𝑤.  (T3) For every 𝑣 ∈ 𝑊 (𝐻), the set 𝑈

𝑣 = {𝑢 ∈ 𝑊 (𝑈) ∶ 𝑣 ∈ 𝑌𝑢}, induces a connected

subtree of T.

Tree Decomposition

 Definition: a tree decomposition of a graph 𝐻 is a pair 𝓤 = (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ), where 𝑈 is a tree whose every node 𝑢 is assigned a vertex subset 𝑌𝑢 ⊆ 𝑊 (𝐻).  The following three conditions hold:

 (T1)ڂ𝑢∈𝑊(𝑈) 𝑌𝑢 = 𝑊(𝐻)  (T2) For every 𝑣𝑤 ∈ 𝐹(𝐻) , there exists a node 𝑢 of 𝑈 such that bag 𝑌𝑢 contains both 𝑣 and 𝑤.  (T3) For every 𝑣 ∈ 𝑊 (𝐻), the set 𝑈

𝑣 = {𝑢 ∈ 𝑊 (𝑈) ∶ 𝑣 ∈ 𝑌𝑢}, induces a connected

subtree of T.

I still didn’t get what is the TREEWIDTH?

Treewidth by a Tree Decomposition

 After we defined what a tree composition is, we can define the treewidth

• f a graph.

 The width of tree decomposition 𝓤 = (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) equals max

𝑢∈𝑊 𝑈 |𝑌𝑢| − 1,

 The treewidth of a graph 𝐻, denoted by 𝑢𝑥(𝐻), is the minimum possible width of a tree decomposition of 𝐻.

Tree Decomposition Intro to Lemma 1.

 Definition 1: (𝐵, 𝐶) is a separation of a graph 𝐻 if 𝐵 ∪ 𝐶 = 𝑊(𝐻) and there is no edge between 𝐵 \ 𝐶 and 𝐶 \ 𝐵  Definition 2: Let (𝐵, 𝐶) be a separation of a graph, then 𝐵 ∩ 𝐶 is a separator

• f this separation, and |𝐵 ∩ 𝐶| is the order of the separation.

({A,B,C,D,E}, {B,E,F,G,H}) is a separation of the graph. {B,E} is the separator

Tree Decomposition Intro to Lemma 1.

 Definition 3: Let A be a subset of 𝑊(𝐻), the border of A, denoted by ∂(A), is the set of those vertices of A that have a neighbor in 𝑊(𝐻)\A.  For us the most crucial property of tree decompositions is that they define a sequence of separators in the graph. For the subset {A,B,C,D,E}, ∂(A)={B,E}

Tree Decomposition

 Lemma 1. Let (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) be a tree decomposition of a graph 𝐻 and let 𝑏𝑐 be an edge of 𝑈. The forest 𝑈 − 𝑏𝑐 obtained from 𝑈 by deleting edge ab consists of two connected components 𝑈

𝑏 (containing a) and 𝑈𝑐 (containing b).

Let K =ڂ𝑢∈𝑊 (𝑈

𝑏) 𝑌𝑢 and M =ڂ𝑢∈𝑊 (𝑈𝑐) 𝑌𝑢.

Then 𝜖(𝐿), 𝜖(𝑁) ⊆ 𝑌𝑏 ∩ 𝑌𝑐. Equivalently, (K, M) is a separation of G with separator 𝑌𝑏 ∩ 𝑌𝑐. 𝑏𝑐 ∈ 𝐹(𝑈)

Tree Decomposition

 Lemma 1. Let (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) be a tree decomposition of a graph 𝐻 and let 𝑏𝑐 be an edge of 𝑈. The forest 𝑈 − 𝑏𝑐 obtained from 𝑈 by deleting edge ab consists of two connected components 𝑈

𝑏 (containing a) and 𝑈𝑐 (containing b).

Let K =ڂ𝑢∈𝑊 (𝑈

𝑏) 𝑌𝑢 and M =ڂ𝑢∈𝑊 (𝑈𝑐) 𝑌𝑢.

Then 𝜖(𝐿), 𝜖(𝑁) ⊆ 𝑌𝑏 ∩ 𝑌𝑐. Equivalently, (K, M) is a separation of G with separator 𝑌𝑏 ∩ 𝑌𝑐.

Tree Decomposition

 Lemma 1. Let (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) be a tree decomposition of a graph 𝐻 and let 𝑏𝑐 be an edge of 𝑈. The forest 𝑈 − 𝑏𝑐 obtained from 𝑈 by deleting edge ab consists of two connected components 𝑈

𝑏 (containing a) and 𝑈𝑐 (containing b).

Let K =ڂ𝑢∈𝑊 (𝑈

𝑏) 𝑌𝑢 and M =ڂ𝑢∈𝑊 (𝑈𝑐) 𝑌𝑢.

Then 𝜖(𝐿), 𝜖(𝑁) ⊆ 𝑌𝑏 ∩ 𝑌𝑐. Equivalently, (K, M) is a separation of G with separator 𝑌𝑏 ∩ 𝑌𝑐. K= {A,B,C,D,E} M= {B,E,F,G,H}

Tree Decomposition

 Lemma 1. Let (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) be a tree decomposition of a graph 𝐻 and let 𝑏𝑐 be an edge of 𝑈. The forest 𝑈 − 𝑏𝑐 obtained from 𝑈 by deleting edge ab consists of two connected components 𝑈

𝑏 (containing a) and 𝑈𝑐 (containing b).

Let K =ڂ𝑢∈𝑊 (𝑈

𝑏) 𝑌𝑢 and M =ڂ𝑢∈𝑊 (𝑈𝑐) 𝑌𝑢.

Then 𝜖(𝐿), 𝜖(𝑁) ⊆ 𝑌𝑏 ∩ 𝑌𝑐. Equivalently, (K, M) is a separation of G with separator 𝑌𝑏 ∩ 𝑌𝑐. K= {A,B,C,D,E} M= {B,E,F,G,H} 𝑌𝑏= {C,B,E} 𝑌𝑐= {B,E,G} 𝑌𝑏 ∩ 𝑌𝑐 = {B,E} 𝜖(𝐿) = {B,E} 𝜖(𝑁) = {B,E}

Tree Decomposition

 Lemma 1. Let (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) be a tree decomposition of a graph 𝐻 and let 𝑏𝑐 be an edge of 𝑈. The forest 𝑈 − 𝑏𝑐 obtained from 𝑈 by deleting edge ab consists of two connected components 𝑈

𝑏 (containing a) and 𝑈𝑐 (containing b).

Let K =ڂ𝑢∈𝑊 (𝑈

𝑏) 𝑌𝑢 and M =ڂ𝑢∈𝑊 (𝑈𝑐) 𝑌𝑢.

Then 𝜖(𝐿), 𝜖(𝑁) ⊆ 𝑌𝑏 ∩ 𝑌𝑐. Equivalently, (K, M) is a separation of G with separator 𝑌𝑏 ∩ 𝑌𝑐. K= {A,B,C,D,E} M= {B,E,F,G,H} 𝑌𝑏= {C,B,E} 𝑌𝑐= {B,E,G} 𝑌𝑏 ∩ 𝑌𝑐 = {B,E} 𝜖(𝐿) = {B,E} 𝜖(𝑁) = {B,E}

Wouldn’t it be complicated to plan the dynamic programming on TD?

Nice Tree Decomposition - motivation

Nice Tree Decomposition

 We will think of a nice tree decompositions as rooted trees.  A (rooted) tree decomposition (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) is nice if the following conditions are satisfied:

 𝑌𝑠 = ∅ for 𝑠 the root of 𝑈 and 𝑌𝑚 = ∅ for every leaf 𝑚 of 𝑈.  Every non-leaf node of 𝑈 is of one of the following three types:

Nice Tree Decomposition

 We will think of a nice tree decompositions as rooted trees.  A (rooted) tree decomposition (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) is nice if the following conditions are satisfied:

 𝑌𝑠 = ∅ for 𝑠 the root of 𝑈 and 𝑌𝑚 = ∅ for every leaf 𝑚 of 𝑈.  Every non-leaf node of 𝑈 is of one of the following three types: 1. Introduce node: a node 𝑢 with exactly one child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢′ ∪ {𝑤} for some vertex 𝑤 ∉ 𝑌𝑢′ (we say that 𝑤 is introduced at 𝑢).

ABC AB Introduce node:

Nice Tree Decomposition

 We will think of a nice tree decompositions as rooted trees.  A (rooted) tree decomposition (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) is nice if the following conditions are satisfied:

 𝑌𝑠 = ∅ for 𝑠 the root of 𝑈 and 𝑌𝑚 = ∅ for every leaf 𝑚 of 𝑈.  Every non-leaf node of 𝑈 is of one of the following three types: 1. Introduce node: a node 𝑢 with exactly one child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢′ ∪ {𝑤} for some vertex 𝑤 ∉ 𝑌𝑢′ (we say that 𝑤 is introduced at 𝑢). 2. Forget node: a node 𝑢 with exactly one child 𝑢′ such that 𝑌𝑢 = Xt

′ ∪ {w} for some vertex

(we say that 𝑥 is forgotten at 𝑢).

AB ABC Forget node:

Nice Tree Decomposition

 We will think of a nice tree decompositions as rooted trees.  A (rooted) tree decomposition (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) is nice if the following conditions are satisfied:

 𝑌𝑠 = ∅ for 𝑠 the root of 𝑈 and 𝑌𝑚 = ∅ for every leaf 𝑚 of 𝑈.  Every non-leaf node of 𝑈 is of one of the following three types: 1. Introduce node: a node 𝑢 with exactly one child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢′ ∪ {𝑤} for some vertex 𝑤 ∉ 𝑌𝑢′ (we say that 𝑤 is introduced at 𝑢). 2. Forget node: a node 𝑢 with exactly one child 𝑢′ such that 𝑌𝑢 = Xt

′ ∪ {w} for some vertex

(we say that 𝑥 is forgotten at 𝑢). 3. Join node: a node 𝑢 with two children 𝑢1, 𝑢2 such that 𝑌𝑢 = 𝑌𝑢1 = 𝑌𝑢2 .

Join node: ABC ABC ABC

Nice Tree Decomposition

 We will think of a nice tree decompositions as rooted trees.  A (rooted) tree decomposition (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) is nice if the following conditions are satisfied:

 𝑌𝑠 = ∅ for 𝑠 the root of 𝑈 and 𝑌𝑚 = ∅ for every leaf 𝑚 of 𝑈.  Every non-leaf node of 𝑈 is of one of the following three types: 1. Introduce node: a node 𝑢 with exactly one child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢′ ∪ {𝑤} for some vertex 𝑤 ∉ 𝑌𝑢′ (we say that 𝑤 is introduced at 𝑢). 2. Forget node: a node 𝑢 with exactly one child 𝑢′ such that 𝑌𝑢 = Xt

′ ∪ {w} for some vertex

(we say that 𝑥 is forgotten at 𝑢). 3. Join node: a node 𝑢 with two children 𝑢1, 𝑢2 such that 𝑌𝑢 = 𝑌𝑢1 = 𝑌𝑢2 . 4. Introduce edge node*: a node 𝑢, labeled with an edge 𝑣𝑤 ∈ 𝐹(𝐻) such that 𝑣, 𝑤 ∈ 𝑌𝑢, and with exactly

• ne child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢

′ . (We say that edge 𝑣𝑤 is introduced at 𝑢).

Edge node: ABC ABC

{A->C}

Nice Tree Decomposition

 We will think of a nice tree decompositions as rooted trees.  A (rooted) tree decomposition (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) is nice if the following conditions are satisfied:

 𝑌𝑠 = ∅ for 𝑠 the root of 𝑈 and 𝑌𝑚 = ∅ for every leaf 𝑚 of 𝑈.  Every non-leaf node of 𝑈 is of one of the following three types: 1. Introduce node: a node 𝑢 with exactly one child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢′ ∪ {𝑤} for some vertex 𝑤 ∉ 𝑌𝑢′ (we say that 𝑤 is introduced at 𝑢). 2. Forget node: a node 𝑢 with exactly one child 𝑢′ such that 𝑌𝑢 = Xt

′ ∪ {w} for some vertex

(we say that 𝑥 is forgotten at 𝑢). 3. Join node: a node 𝑢 with two children 𝑢1, 𝑢2 such that 𝑌𝑢 = 𝑌𝑢1 = 𝑌𝑢2 . 4. Introduce edge node*: a node 𝑢, labeled with an edge 𝑣𝑤 ∈ 𝐹(𝐻) such that 𝑣, 𝑤 ∈ 𝑌𝑢, and with exactly

• ne child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢

′ . (We say that edge 𝑣𝑤 is introduced at 𝑢).

Edge node: ABC ABC

{A->C}

Isn’t the nice tree decomposition width bigger then the TW of the graph?

Nice Tree Decomposition

 Lemma 2. If a graph 𝐻 admits a tree decomposition of width at most 𝑙, then it also admits a nice tree decomposition of width at most 𝑙.  Moreover, given a tree decomposition 𝓤 = (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) of 𝐻 of width at most 𝑙, one can in time 𝑃(𝑙2 · max(|𝑊 (𝑈)|, |𝑊 (𝐻)|)) compute a nice tree decomposition of G of width at most k that has at most 𝑃(𝑙|𝑊(𝐻)|) nodes.  Proof of the lemma and the algorithm of computing a (nice) tree decomposition are out of the scope of this lecture. Therefore, we will assume that such a decomposition is provided on the input together with the graph.

Dynamic Programming Reminder

 Dynamic Programming(DP) is a technique to solve problems by breaking them down into overlapping sub-problems which follows the optimal substructure.  Dynamic programming is a class of problems where it is possible to store results for recurring computations in some lookup so that they can be used when required again by other computations.  This improves performance at the cost of memory.  We will focus on dynamic programming for graphs of bounded treewidth

Dynamic Programming Reminder

 Dynamic Programming(DP) is a technique to solve problems by breaking them down into overlapping sub-problems which follows the optimal substructure.  Dynamic programming is a class of problems where it is possible to store results for recurring computations in some lookup so that they can be used when required again by other computations.  This improves performance at the cost of memory.  We will focus on dynamic programming on graphs of bounded treewidth

Which graphs are

• f bounded

treewidth?

Dynamic Programming on graphs of bounded treewidth

 Examples of graphs with bounded treewidth:  Pseudoforest graph: every connected component has at most one cycle. 𝑢𝑠𝑓𝑓𝑥𝑗𝑒𝑢ℎ = 2

Dynamic Programming on graphs of bounded treewidth

 Examples of graphs with bounded treewidth:  Cactus graph: any two simple cycles have at most

• ne vertex in common

𝑢𝑠𝑓𝑓𝑥𝑗𝑒𝑢ℎ = 2

Dynamic Programming on graphs of bounded treewidth

 Examples of graphs with bounded treewidth:  outerplanar graph: has a planar drawing for which all vertices belong to the outer face

• f the drawing.

𝑢𝑠𝑓𝑓𝑥𝑗𝑒𝑢ℎ = 2

Dynamic Programming on graphs of bounded treewidth

 Examples of graphs with bounded treewidth:  Control flow graph (compilation): all paths that might be traversed through a program during its execution. 𝑢𝑠𝑓𝑓𝑥𝑗𝑒𝑢ℎ ≤ 6

Maximum Weighted Independent Set

 Independent Set: Given an undirected graph G=(V,E) an independent set (IS) of G is a subset 𝑇 ⊆ 𝑊, such that no two of its vertices are adjacent.  The problem: Given an undirected graph G=(V,E) and a weight function on its vertices 𝑥: 𝑊 → ℝ+, find a subset 𝑇 ⊆ 𝑊 such that 𝑇 ∈ 𝐽𝑇 and ∀𝑇′ ∈ 𝐽𝑇 ∶ 𝑥 𝑇 ≥ 𝑥(𝑇′).  The maximum weighted independent set is known to be NP-hard.  Therefore, it is unlikely that there exists an efficient algorithm for solving it.  However, we will now see a dynamic-programming-based algorithm that solves it efficiently on graphs of bounded treewidth.

Weighted Independent Set – DP alg.

 Let G=(V,E) be a graph of n-vertex with width of at most k.  And let 𝓤 = (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) be a tree decomposition of G.  By applying Lemma 2 we can assume that 𝓤 is a nice tree decomposition.

Weighted Independent Set – DP alg.

 Let G=(V,E) be a graph of n-vertex with width of at most k.  And let 𝓤 = (𝑈, 𝑌𝑢 𝑢∈𝑊 𝑈 ) be a tree decomposition of G.  By applying Lemma 2 we can assume that 𝓤 is a nice tree decomposition.

Weighted Independent Set – DP alg intro.

 Recall that 𝑈 is rooted at some node 𝑠.  For a node 𝑢 of 𝑈, let 𝑊

𝑢 be the union of all

the bags present in the subtree of 𝑈 rooted at 𝑢, including 𝑌𝑢.  Provided that 𝑢 ≠ 𝑠 we can apply Lemma 1 to the edge of 𝑈 between 𝑢 and its parent, and infer that 𝜖 𝑊

𝑢 ⊆ 𝑌𝑢 .

 The same conclusion is trivial when 𝑢 = 𝑠. 𝑢 𝑊

𝑢

Weighted Independent Set – DP alg intro.

 Recall that 𝑈 is rooted at some node 𝑠.  For a node 𝑢 of 𝑈, let 𝑊

𝑢 be the union of all

the bags present in the subtree of 𝑈 rooted at 𝑢, including 𝑌𝑢.  Provided that 𝑢 ≠ 𝑠 we can apply Lemma 1 to the edge of 𝑈 between 𝑢 and its parent, and infer that 𝜖 𝑊

𝑢 ⊆ 𝑌𝑢 .

 The same conclusion is trivial when 𝑢 = 𝑠. 𝑢 𝑊

𝑢

Lemma 1. K =ڂ𝑢∈𝑊 (𝑈

𝑏) 𝑌𝑢,

and M =ڂ𝑢∈𝑊 (𝑈𝑐) 𝑌𝑢 , (K, M) is a separation of G with separator 𝑌𝑏 ∩ 𝑌𝑐.

Weighted Independent Set – DP alg.

 Among independent sets 𝐽 satisfying 𝐽 ∩ 𝑌𝑢 = 𝑇 for some fixed 𝑇, all the maximum-weight solutions have exactly the same weight of the part contained in 𝑊

𝑢.

Weighted Independent Set – DP alg.

 Among independent sets 𝐽 satisfying 𝐽 ∩ 𝑌𝑢 = 𝑇 for some fixed 𝑇, all the maximum-weight solutions have exactly the same weight of the part contained in 𝑊

𝑢.

 For every node 𝑢 and every 𝑇 ⊆ 𝑌𝑢, define the following value: 𝑑 𝑢, 𝑇 = 𝑛𝑏𝑦𝑗𝑛𝑣𝑛 𝑞𝑝𝑡𝑡𝑗𝑐𝑚𝑓 𝑥𝑓𝑗𝑕ℎ𝑢 𝑝𝑔 𝑏 𝑡𝑓𝑢 መ 𝑇 𝑡𝑣𝑑ℎ 𝑢ℎ𝑏𝑢 𝑇 ⊆ መ 𝑇 ⊆ 𝑊

𝑢, መ

𝑇 ∩ 𝑌𝑢 = 𝑇, 𝑏𝑜𝑒 መ 𝑇 𝑗𝑡 𝑗𝑜𝑒𝑓𝑞𝑓𝑜𝑒𝑓𝑜𝑢.

Weighted Independent Set – DP alg.

 Among independent sets 𝐽 satisfying 𝐽 ∩ 𝑌𝑢 = 𝑇 for some fixed 𝑇, all the maximum-weight solutions have exactly the same weight of the part contained in 𝑊

𝑢.

 For every node 𝑢 and every 𝑇 ⊆ 𝑌𝑢, define the following value: 𝑑 𝑢, 𝑇 = 𝑛𝑏𝑦𝑗𝑛𝑣𝑛 𝑞𝑝𝑡𝑡𝑗𝑐𝑚𝑓 𝑥𝑓𝑗𝑕ℎ𝑢 𝑝𝑔 𝑏 𝑡𝑓𝑢 መ 𝑇 𝑡𝑣𝑑ℎ 𝑢ℎ𝑏𝑢 𝑇 ⊆ መ 𝑇 ⊆ 𝑊

𝑢, መ

𝑇 ∩ 𝑌𝑢 = 𝑇, 𝑏𝑜𝑒 መ 𝑇 𝑗𝑡 𝑗𝑜𝑒𝑓𝑞𝑓𝑜𝑒𝑓𝑜𝑢.  If no such set መ 𝑇 exists, then we put 𝑑[𝑢, 𝑇] = −∞ (iff S is not independent)  Final solution is 𝑑[𝑠, ∅].

Weighted Independent Set – DP alg.

 Among independent sets 𝐽 satisfying 𝐽 ∩ 𝑌𝑢 = 𝑇 for some fixed 𝑇, all the maximum-weight solutions have exactly the same weight of the part contained in 𝑊

𝑢.

 For every node 𝑢 and every 𝑇 ⊆ 𝑌𝑢, define the following value: 𝑑 𝑢, 𝑇 = 𝑛𝑏𝑦𝑗𝑛𝑣𝑛 𝑞𝑝𝑡𝑡𝑗𝑐𝑚𝑓 𝑥𝑓𝑗𝑕ℎ𝑢 𝑝𝑔 𝑏 𝑡𝑓𝑢 መ 𝑇 𝑡𝑣𝑑ℎ 𝑢ℎ𝑏𝑢 𝑇 ⊆ መ 𝑇 ⊆ 𝑊

𝑢, መ

𝑇 ∩ 𝑌𝑢 = 𝑇, 𝑏𝑜𝑒 መ 𝑇 𝑗𝑡 𝑗𝑜𝑒𝑓𝑞𝑓𝑜𝑒𝑓𝑜𝑢.  If no such set መ 𝑇 exists, then we put 𝑑[𝑢, 𝑇] = −∞ (iff S is not independent)  Final solution is 𝑑[𝑠, ∅].  Now all we have to do is defining our recursive formulas for bottom-up DP.

Weighted Independent Set – DP alg.

 Thanks to the definition of nice tree decomposition we have only few cases of how a bag relates to its children.  The computation 𝐷[𝑢, 𝑇] for each node is based

• nly on the values that were computed already

for the children of this node.  Leaf node. If t is a leaf node, then we have only

• ne value 𝑑[𝑢, ∅] = 0.

Leaf node

Weighted Independent Set – DP alg.

 Introduce node. Suppose 𝑢 is an introduce node with child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢

′ ∪ {𝑤} for some 𝑤 ∉ 𝑌𝑢 ′ .

Let 𝑇 be any subset of 𝑌𝑢. If 𝑇 is not independent, then we can immediately put 𝑑[𝑢, 𝑇] = −∞; hence assume

• therwise.

Then we claim that the following formula holds: Introduce node *

Weighted Independent Set – DP alg.

 Proof of *:  𝑤 ∈ 𝑇. Let መ

𝑇 be the set that maximize 𝑑[𝑢, 𝑇].  መ 𝑇 ∖ {𝑤} was considered in the calculation of 𝑑[𝑢′, 𝑇 ∖ {𝑤}]

(no other vertices were added, here nice tree helps us)  => 𝑑[𝑢′, 𝑇 ∖ {𝑤}] ≥ 𝑥( መ

𝑇 ∖ {𝑤}) = 𝑥( መ 𝑇) − 𝑥(𝑤) = 𝑑[𝑢, 𝑇] − 𝑥(𝑤).

 => *

Weighted Independent Set – DP alg.

 Proof of * (cont.):  𝑤 ∈ 𝑇. Let ෡ 𝑇′ be the set that maximize 𝑑[𝑢′, 𝑇 ∖ {𝑤}].  𝑇 is independent (as we assume before) so 𝑤 doesn’t have neighbors in S ∖ {𝑤} = ෡ 𝑇′ ∩ 𝑌𝑢′  Moreover, by lemma 1 , 𝑤 doesn’t have any neighbor in 𝑊

𝑢 ′ ∖ 𝑌𝑢′ ⊇ ෡

𝑇′ ∖ 𝑌𝑢′  => 𝑤 Doesn’t have neighbor in ෡ 𝑇′  => ෡ 𝑇′ڂ{𝑤} is independent Set.  ෡ 𝑇′ڂ{𝑤} intersects with 𝑌𝑢 only at 𝑇 so this set was consider for 𝑑[𝑢, 𝑇]. *

Weighted Independent Set – DP alg.

 Proof of * (cont.):  Now we can conclude:  And from both conclusions we get: *

Weighted Independent Set – DP alg.

 Forget node. Suppose 𝑢 is a forget node with child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢′ ∖ {𝑥} for some 𝑥 ∈ 𝑌𝑢 . Let 𝑇 be any subset of 𝑌𝑢 ; again we assume that S is independent, since otherwise we put c[t, S] = −∞. We claim that the following formula holds:  Proof: Let ෡ 𝑇′ be a set that maximize 𝑑[𝑢, 𝑇]. If 𝑥 ∉ ෡ 𝑇′, so ෡ 𝑇′ was considered when calculating 𝑑[𝑢’, 𝑇] and hence 𝑑[𝑢’, 𝑇] ≥ 𝑥(෡ 𝑇′) = 𝑑[𝑢, 𝑇]. Else, If 𝑥 ∈ ෡ 𝑇′ so ෡ 𝑇′ was considered when calculating 𝑑[𝑢’, 𝑇 ∪ {𝑥}] and hence 𝑑[𝑢’, 𝑇 ∪ {𝑥}]] ≥ 𝑥(෡ 𝑇′) = 𝑑[𝑢, 𝑇]. Forget node

Weighted Independent Set – DP alg.

 Forget node. Suppose 𝑢 is a forget node with child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢′ ∖ {𝑥} for some 𝑥 ∈ 𝑌𝑢 . Let 𝑇 be any subset of 𝑌𝑢 ; again we assume that S is independent, since otherwise we put c[t, S] = −∞. We claim that the following formula holds: Forget node

Weighted Independent Set – DP alg.

 Forget node. Suppose 𝑢 is a forget node with child 𝑢′ such that 𝑌𝑢 = 𝑌𝑢′ ∖ {𝑥} for some 𝑥 ∈ 𝑌𝑢 . Let 𝑇 be any subset of 𝑌𝑢 ; again we assume that S is independent, since otherwise we put c[t, S] = −∞. We claim that the following formula holds: Forget node

Weighted Independent Set – DP alg.

 Join node. Finally, suppose that 𝑢 is a join node with children 𝑢1, 𝑢2 such that 𝑌𝑢 = 𝑌𝑢1 = 𝑌𝑢2 . Let 𝑇 be any subset of 𝑌𝑢; as before, we can assume that 𝑇 is independent. The recursive formula is as follows:  Proof idea: from Lemma 1 we know that each part of 𝑢’s children is separated and the border is inside 𝑌𝑢

• vertices. We take all of 𝑇(⊆ 𝑌𝑢) to be in 𝑑 𝑢, 𝑇 . So we

will consider the best solution of each child for S and subtract its size once (because we took it twice). Join node

Run-time analysis of the algorithm

 We have treewidth of at most 𝑙, which means 𝑌𝑢 ≤ 𝑙 + 1 for every node 𝑢.  Thus for every node 𝑢 we compute 2 𝑌𝑢 ≤ 2𝑙+1 values of 𝑑[𝑢, 𝑇]  In naive solution we will say that each 𝑑[𝑢, 𝑇] computed in 𝑜𝑃 1 time. It is possible to construct a data structure that allows performing adjacency queries in time 𝑃 𝑙 , so computing each 𝑑 𝑢, 𝑇 will take only 𝑙𝑃 1 time.  we assumed that the number of nodes of the given tree decompositions is 𝑃(𝑙𝑜) (Lemma 2)  the total running time of the algorithm is 2𝑙 · 𝑙𝑃 1 · 𝑜

We’ve got a FPT algorithm for a problem that is known to be NP-hard!

Run-time analysis of the algorithm

 We have treewidth of at most 𝑙, which means 𝑌𝑢 ≤ 𝑙 + 1 for every node 𝑢.  Thus for every node 𝑢 we compute 2 𝑌𝑢 ≤ 2𝑙+1 values of 𝑑[𝑢, 𝑇]  In naive solution we will say that each 𝑑[𝑢, 𝑇] computed in 𝑜𝑃 1 time. It is possible to construct a data structure that allows performing adjacency queries in time 𝑃 𝑙 , so computing each 𝑑 𝑢, 𝑇 will take only 𝑙𝑃 1 time.  we assumed that the number of nodes of the given tree decompositions is 𝑃(𝑙𝑜) (Lemma 2)  the total running time of the algorithm is 2𝑙 · 𝑙𝑃 1 · 𝑜

We’ve got a FPT algorithm for a problem that is known to be NP-hard! Just for graphs of bounded treewidth though

Monadic second-order Logic

 In the course of Logic we saw First-Order Logic, where we can use quantifiers only

• ver variables that range over individuals (∀𝑦 𝑏𝑜𝑒 ∃𝑦).

∀𝑦∀𝑧 𝑆 𝑦, 𝑧 → ¬𝑆 𝑧, 𝑦 (𝑏 − 𝑡𝑧𝑛𝑛𝑓𝑢𝑠𝑗𝑑 𝑒𝑓𝑔. )  second-order logic is an extension of it, that allows us to use quantifiers over relations, functions and sets of elements.  monadic second order logic (𝑵𝑻𝑷) is the fragment of second-order logic where the second-order quantification is limited to be only over sets.  𝑵𝑻𝑷𝟑 allows quantification over sets of vertices or edges.  Our main interest here is to use MSO to describe properties of undirected graphs.  We view an undirected graph as relational structure (i.e. a model as in logic), where the universe is the vertices and there is one binary relation 𝐹(𝑦, 𝑧) for the edges.  Expmle for such 𝑵𝑻𝑷𝟑 formula that express 3-coloring in a graph:

∃𝑌1∃𝑌2∃𝑌3 ∀𝑦 ሧ

𝑗

𝑌𝑗 ∧ ∀𝑦∀𝑧 𝐹 𝑦, 𝑧 → ሧ

𝑗≠𝑘

𝑌𝑗 𝑦 ∧ 𝑌

𝑘(𝑦)

Courcelle's theorem

 Courcelle's theorem. Assume that 𝜚 is a formula of 𝑁𝑇𝑃2 and 𝐻 is an n- vertex graph. Suppose, moreover, that a tree decomposition of 𝐻 of width 𝑢 is provided. Then there exists an algorithm that verifies whether ϕ is satisfied in G in time 𝑔(||𝜚||, 𝑢) · 𝑜, for some computable function f.  In other words, every graph property definable in the monadic second-

• rder logic of graphs can be decided in linear time on graphs of

bounded treewidth.

Courcelle's theorem

 Courcelle's theorem. Assume that 𝜚 is a formula of 𝑁𝑇𝑃2 and 𝐻 is an n- vertex graph. Suppose, moreover, that a tree decomposition of 𝐻 of width 𝑢 is provided. Then there exists an algorithm that verifies whether ϕ is satisfied in G in time 𝑔(||𝜚||, 𝑢) · 𝑜, for some computable function f.  In other words, every graph property definable in the monadic second-

• rder logic of graphs can be decided in linear time on graphs of

bounded treewidth. So which more problems are tractable now?

Courcelle's theorem

 Courcelle's theorem. Assume that 𝜚 is a formula of 𝑁𝑇𝑃2 and 𝐻 is an n- vertex graph. Suppose, moreover, that a tree decomposition of 𝐻 of width 𝑢 is provided. Then there exists an algorithm that verifies whether ϕ is satisfied in G in time 𝑔(||𝜚||, 𝑢) · 𝑜, for some computable function f.  In other words, every graph property definable in the monadic second-

• rder logic of graphs can be decided in linear time on graphs of

bounded treewidth. So which more problems are tractable now?

That’s all, Thank you for listening

Any questions?