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Treewidth Approximation

Treewidth is a measure of how similar a graph is to a tree. It has many algorithmic applications.

Overview

Treewidth measures how close a graph is to being a tree. Computing exact treewidth is NP-hard, but approximation algorithms exist. PyGraphina provides two heuristic-based approximation algorithms: min-degree and min-fill-in.

Function Signatures

pg.approximation.treewidth_min_degree(graph: PyGraph) -> Tuple[int, List[int]]
pg.approximation.treewidth_min_fill_in(graph: PyGraph) -> Tuple[int, List[int]]

Parameters

  • graph: The undirected graph to analyze

Returns

A tuple containing: - int: Upper bound on treewidth - List[int]: Elimination ordering of nodes

Algorithms

Min-Degree Heuristic

Iteratively removes the node with minimum degree. Fast but can produce suboptimal results.

Time Complexity: O(n³) with simple implementation
Quality: Often reasonable, sometimes suboptimal

Min-Fill-In Heuristic

Removes the node that creates the fewest new edges when eliminated. Slower but often produces better bounds.

Time Complexity: O(n⁴) with simple implementation
Quality: Generally better than min-degree

Time Complexity

  • Min-Degree: O(n³)
  • Min-Fill-In: O(n⁴)

Where n is the number of nodes.

Use Cases

  • Dynamic programming on graphs
  • SAT/constraint satisfaction solving
  • Database query optimization
  • Graph coloring approximation
  • Temporal reasoning in AI
  • Network design and analysis

Example

import pygraphina as pg

# Create a graph
g = pg.PyGraph()
nodes = [g.add_node(i) for i in range(10)]

# Tree-like structure
for i in range(9):
    g.add_edge(nodes[i], nodes[i + 1], 1.0)

# Add some edges to increase treewidth
g.add_edge(nodes[0], nodes[2], 1.0)
g.add_edge(nodes[0], nodes[3], 1.0)
g.add_edge(nodes[1], nodes[3], 1.0)

# Compare algorithms
width_md, order_md = pg.approximation.treewidth_min_degree(g)
width_mfi, order_mfi = pg.approximation.treewidth_min_fill_in(g)

print(f"Min-Degree Treewidth: {width_md}")
print(f"Min-Fill-In Treewidth: {width_mfi}")
print(f"Elimination order (min-degree): {order_md}")
print(f"Elimination order (min-fill-in): {order_mfi}")

# A tree has treewidth 1
# Adding more edges increases treewidth
tree = pg.PyGraph()
tree_nodes = [tree.add_node(i) for i in range(5)]
for i in range(4):
    tree.add_edge(tree_nodes[i], tree_nodes[i+1], 1.0)

tree_width, _ = pg.approximation.treewidth_min_degree(tree)
print(f"\nTreewidth of a path (tree): {tree_width}")

Interpreting Results

  • Treewidth = 1: Graph is a tree or forest
  • Treewidth = 2: Graph is outerplanar
  • Treewidth = k: Graph has tree decomposition of width k
  • Lower is better: Enables faster algorithms

Significance

Low treewidth enables:

  • Efficient exact algorithms for NP-hard problems
  • Decomposition into tree-structured subproblems
  • Polynomial algorithms for many hard problems on graphs with bounded treewidth
  • Better understanding of graph structure

Algorithm Selection

  • Use Min-Degree: When you need fast approximation, large graphs
  • Use Min-Fill-In: When you need better bounds, smaller/medium graphs
  • Compare both: For important applications to pick the better result

Applications

Application Why Treewidth Matters
SAT Solving Low treewidth → faster solving
Database Queries Optimization on tree decomposition
Probabilistic Inference Belief propagation on trees
Scheduling Decompose into subproblems
Network Analysis Understand structure complexity