That's definitely frustrating. I've been in the same boat a number of times and had decent success with a few tactics. Before you start looking at Negative SEO techniques or responding with underhanded tactics, first look at your brand.
Do you feel your brand is stronger and more solid than the copying brand?
Do you have something that's unique that they are unable to copy?
If the answer is yes to both of those questions, then first focus on improving your service and look to make your brand stand out a bit more from the copy. As your brand grows stronger, their brand will get pushed back naturally. Start pushing on social media and engaging your followers to post, comment, and share your posts. This engagement will boost your SEO and the positive comments from your audience will start to show that you offer a better product.
That's the long term solution. In the short term, look and see what they're copying. If they're copying your posts word for word, then you can report that to Google. Google has a Report Copyright page for this:
https://www.google.com/webmasters/tools/dmca-notice?pli=1&
Google gets to these in time and will eventually drop their pages. You can also DMCA their hosting company if they're clearly copying your articles.
To combat any false negative articles they're writing about you, I would start leveraging social media. If you can get your clientele to start engaging your page, post good comments, and like posts, users who search for your brand will see that you offer a legitimate service. Follow up by writing press releases that link to your social media pages, articles, guest posts, and get the word out that you have fans who've left great feedback.
If you can successfully show that you're better without getting your hands dirty, your brand will be much stronger than the copy. They will eventually fall behind and be unable to keep up. The down side is it takes a bit more effort. However, I've taken the low road before and the problem is you end up just bickering at the same level. Neither company moves ahead because all they do is fight rather than improve their products.