Welcome to Goy Dinah. Presumably this is growing in your neighbours garden. What is it growing on eg trellis, fence, tree or ?. It would be best if you can get them to cut it back. Or are the neighbours complaining about a plant you own? If you want to get rid of it completely cut the stems as close to the ground as possible and put Ivy killer on the cut ends as soon as you cut them. That should kill the roots. It may need more than one go at it, if it is very rampant. The bits above the ground should die from lack of moisture.

Or if you want to be sly, copper nails. Not that I would advocate such a thing.....

Yes, it's not clear whether you mean your neighbours next door are unhappy about your overgrown ivy, or whether you're unhappy about theirs...

If the ivy belongs to your neighbour you could just keep it close and neat to your side of the fence . Ivy makes a good winter shelter for many harmless creatures and nesting places for wrens and robins. It is important to the holly-blue butterfly and several types of bee. You can also grow a colourful annual such climbing nasturtiums up it for the summer.
(This is what I do with the ivy that smothers the fence at the bottom of my garden and it seems to work quite well).