# On the Complexity of Local Graph Transformations

## TL;DR

This paper studies the complexity of transforming one graph into another using local, node-based primitives, proving NP-hardness for optimal sequences and providing a polynomial-time approximation algorithm.

## Contribution

It introduces a formal framework for local graph transformations, proves NP-hardness of finding minimal transformation sequences, and offers an efficient approximation algorithm.

## Key findings

- Transforming graphs with local primitives is NP-hard.
- A polynomial-time approximation algorithm with a constant ratio exists.
- Local primitives cannot disconnect or add nodes to the graph.

## Abstract

We consider the problem of transforming a given graph $G_s$ into a desired graph $G_t$ by applying a minimum number primitives from a particular set of local graph transformation primitives. These primitives are local in the sense that each node can apply them based on local knowledge and by affecting only its $1$-neighborhood. Although the specific set of primitives we consider makes it possible to transform any (weakly) connected graph into any other (weakly) connected graph consisting of the same nodes, they cannot disconnect the graph or introduce new nodes into the graph, making them ideal in the context of supervised overlay network transformations. We prove that computing a minimum sequence of primitive applications (even centralized) for arbitrary $G_s$ and $G_t$ is NP-hard, which we conjecture to hold for any set of local graph transformation primitives satisfying the aforementioned properties. On the other hand, we show that this problem admits a polynomial time algorithm with a constant approximation ratio.