Graph Theory - Tree Decomposition

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Tree Decomposition

Tree Decomposition is used to map a graph into a tree-like structure. It provides a way to analyze and solve problems on complex graphs by breaking them into simpler components. The main idea is to break the graph into smaller, overlapping subsets of vertices, organized in a tree structure, where each subset represents a "bag".

The above diagram shows how a graph is represented as a tree decomposition, with vertices grouped into overlapping bags connected in a tree structure.

Properties of Tree Decomposition

Following are the important properties of tree decomposition −

• Bag Coverage: Each vertex of the graph must appear in at least one bag of the tree decomposition.
• Edge Coverage: For every edge in the graph, both vertices of the edge must appear together in at least one bag.
• Connected Bags: For any vertex in the graph, the bags containing that vertex must form a connected subtree in the decomposition tree.

Treewidth

Treewidth is a measure of how closely a graph resembles a tree. It is defined as the size of the largest bag in the tree decomposition, minus one.

In other words, it represents how complex the connections are in the graph when viewed through a tree-like decomposition. A graph with treewidth 1 is a tree itself, while higher treewidth values indicate more complex graphs.

The treewidth of a graph is formally defined as:

Treewidth = max(|Bag|) - 1

For example, a tree has a treewidth of 1, while a complete graph with n vertices has a treewidth of n - 1.

Constructing a Tree Decomposition

The process of constructing a tree decomposition involves dividing the graph into overlapping subsets of vertices and arranging them in a tree structure. The steps are −

• Identify subsets of vertices (bags) such that each vertex and edge of the graph is covered.
• Ensure the bags satisfy the properties of bag coverage, edge coverage, and connected bags.
• Connect the bags in a tree structure.

Let us look at an example −

Graph: [(A, B), (A, C), (B, C), (C, D)]

Tree Decomposition:
Bag 1: [A, B, C]
Bag 2: [C, D]

The decomposition ensures that all edges are covered, and bags containing the same vertex (e.g., C) form a connected subtree.

Applications of Tree Decomposition

Tree decomposition is useful in solving various complex problems in graph theory and real-world applications −

• Solving NP-hard Problems: Many NP-hard problems, such as graph coloring and finding Hamiltonian paths, become easier to solve on graphs with bounded treewidth.
• Constraint Satisfaction Problems (CSP): Tree decomposition helps break down CSPs into simpler subproblems, making them easier to solve.
• Database Query Optimization: In databases, tree decomposition helps improve query performance by reducing unnecessary calculations.

Tree Decomposition Algorithms

There are different algorithms used to find tree decompositions, such as −

• Naive Algorithm: A simple method that tries all possible options to create a decomposition.
• Greedy Algorithms: These use a step-by-step approach to build a decomposition by picking vertices or edges based on certain properties.
• Dynamic Programming: A faster method to compute tree decompositions for specific types of graphs, like chordal graphs.

Example: Naive Algorithm

The naive algorithm explores all possible subsets of vertices to construct a tree decomposition. For a small graph, this approach is feasible but inefficient for larger graphs.

Graph: [(A, B), (A, C), (B, C), (C, D)]

Step 1: Identify subsets:
Subset 1: [A, B, C]
Subset 2: [C, D]

Step 2: Connect subsets:
Tree structure: [Subset 1 - Subset 2]