# Common greedy wiring and rewiring heuristics do not guarantee maximum assortative graphs of given degree

## TL;DR

This paper investigates greedy heuristics for constructing maximum assortative graphs with a given degree sequence, revealing their limitations through counterexamples that show they do not always produce optimal solutions.

## Contribution

It demonstrates that common greedy wiring and rewiring heuristics can fail to find maximum assortative graphs, challenging assumptions about their effectiveness.

## Key findings

- Greedy rewiring heuristics do not always produce maximum assortative graphs.
- A greedy wiring heuristic may fail to achieve the target degree sequence.
- Counterexamples illustrate the limitations of these heuristics.

## Abstract

We examine two greedy heuristics - wiring and rewiring - for constructing maximum assortative graphs over all simple connected graphs with a target degree sequence. Counterexamples show that natural greedy rewiring heuristics do not necessarily return a maximum assortative graph, even though it is known that the meta-graph of all simple connected graphs with given degree is connected under rewiring. Counterexamples show an elegant greedy graph wiring heuristic from the literature may fail to achieve the target degree sequence or may fail to wire a maximally assortative graph.