Solving Subgraph Isomorphism in Z3 with Python

Learn how to encode and solve the problem of `subgraph isomorphism` using Z3 and Python. Discover a simple coding approach to check graph mappings effectively!
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This video is based on the question https://stackoverflow.com/q/67493594/ asked by the user 'vputz' ( https://stackoverflow.com/u/5754477/ ) and on the answer https://stackoverflow.com/a/67494282/ provided by the user 'alias' ( https://stackoverflow.com/u/936310/ ) at 'Stack Overflow' website. Thanks to these great users and Stackexchange community for their contributions.

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Understanding and Solving Subgraph Isomorphism in Z3

Subgraph isomorphism is a fundamental problem in graph theory, often explored in computer science, particularly in areas involving network analysis and data structure comparison. The challenge lies in determining if a smaller graph (the subgraph) can be mapped to a portion of a larger graph (the supergraph) while preserving the connectivity of nodes.

In this guide, we will walk through how to encode this problem using Z3 and Python (z3py) with an example that outlines the steps effectively.

Problem Brief

The problem involves determining whether a subgraph can be embedded within a supergraph such that the connected structure is maintained. For instance, consider the following graphs:

Subgraph: Nodes (0, 1, 2) with edge (0, 1)

Supergraph: Nodes (0, 1, 2) with edge (1, 2)

The goal is to find a mapping from the nodes in the subgraph to nodes in the supergraph. One possible mapping could be:

[[See Video to Reveal this Text or Code Snippet]]

The Challenge

While encoding subgraph isomorphism in Z3 might seem straightforward, it quickly becomes complex when working with larger graphs. The goal is to ensure that:

Each node in the subgraph is assigned to a unique node in the supergraph.

If two nodes are connected in the subgraph, their corresponding nodes must also be connected in the supergraph.

Solution Steps

Let's delve into the approach for solving this problem using Z3.

1. Setup Your Environment

Before you start coding, ensure you have z3 installed in your Python environment. You can install it using pip:

[[See Video to Reveal this Text or Code Snippet]]

2. Define the Graphs

Here’s how to set up your subgraph and supergraph in Python using the networkx library:

[[See Video to Reveal this Text or Code Snippet]]

3. Create the Mapping

Next, we will create an array to represent the mapping of nodes from the subgraph to the supergraph:

[[See Video to Reveal this Text or Code Snippet]]

4. Define Constraints

We need to add constraints to ensure:

Each node in the subgraph is mapped to a node in the supergraph.

No two nodes in the subgraph are assigned the same node in the supergraph.

[[See Video to Reveal this Text or Code Snippet]]

5. Preserve Edge Connectivity

The last piece is to ensure that if there's an edge between two nodes in the subgraph, the mapped nodes also maintain that connection in the supergraph:

[[See Video to Reveal this Text or Code Snippet]]

6. Solve the Problem

Finally, you can run the solver and check whether a valid mapping exists:

[[See Video to Reveal this Text or Code Snippet]]

Summary of Output

When executed, the solution might yield a mapping similar to:

[[See Video to Reveal this Text or Code Snippet]]

This confirms that the connections and mappings are valid as per the rules we've set.

Conclusion

In this post, we’ve tackled the problem of subgraph isomorphism using Z3 with Python through a structured approach. While encoding complex graphs may lead to challenges related to scalability, our example demonstrates a foundational understanding of the mechanics involved.

Feel free to use this as a starting point for more complex graph structures or to explore other properties within the Z3 framework!

To implement this solution in your own projects or studies, simply tweak the defined graphs and constraints according to your requirements. Happy coding!