A friend of mine had a similar situation and told me this story.
He had a neighbor who's dog barked all the time constantly and was just as bad as the one you've experienced. He tried being nice and telling the neighbor and asking them to please tie the dog up. They ignored his request and didn't care. It was his experience that some dog owners, (like some parents with obnoxious children in stores), are so used to their obnoxious children or animals, that they expect everyone else to put up with it too and don't care who it bothers.
He told me it got so bad that he called animal control who came out and talked with the dog owner but the dog owner didn't pay any attention to them either. So this friend filed suit against the dog owner for nuisance and noise ordinance and recorded the dog on video/audio. It went to court, my friend played his video/audio recording and the judge made the dog owner pay a fine. This friend then thought that would stop it. But it didn't. They continued doing the same thing even after going to court and paying a fine. According to the story my friend told me, the authorities are loathe to get involved in situations like this and even when he took the time, trouble and expense to file in court and won, it didn't stop the situation.
This friend of mine put his home up for sale and moved because of it not stopping. He told me if he had it to do all over again, he had learned it would be best not to say anything at all to the dog owner. Because then if anything happened to the dog they would know who to retaliate against. He told me if it ever happened to him again, that he wouldn't say anything to the owner (based on his previous experience) but would specially prepare a treat for that dog to eat and toss it to him.
That's the story that was told to me by my friend. I personally would not advocate what he described he would do if it ever happened to him again, but I can understand his frustration at eventually having to move due to the dog owner's lack of controlling his animal. Situations such as this are difficult. My friend told me that some people have been killed over arguments concerning a dog. He said some people consider their dogs to be like their children and that they can do no wrong and get uppity with you if you dare to even politely ask them to control them. He said by asking them to control their animal, you are in effect telling them they are not being responsible, and that people being people, will take offense at that even if you are polite about it.
You could fight fire with fire and put your neighbor in your shoes. Perhaps you could let your dogs out, so that they can take care of that other dog without you doing anything personally. It would help if one of your dogs was a big bad Rottweiler or similar large dog and with a nasty disposition that could eat the other dog in one bite. Perhaps that would intimidate the other dog to the point where he dares not utter a peep and stays very quiet and hidden for fear of being pounced upon by your dogs. Then that puts the other dog owner in your shoes. Now HE would have to come to you asking you to control your animal. Perhaps then the two of you could come to an equitable conciliation.
Or you could simply let that persons yap yap dog bite your ankle next time he tries to bite you (how bad could it be with a tiny yap yap dog?). Then immediately call 911 and animal control and go to the doctor. Once the dog bites you, the authorities will likely make the dog be put down. If they don't, you can sue the owner for his dog biting you and your pain and suffering and medical expenses. I imagine that would also put an end to the problem.
Or...if the dog comes on your property, put a box propped up with a stick tied to a piece of string with a piece of meat under the box. When the dog goes for the meat, pull the stick out and capture the dog. Or go to the expense of using a wire dog capture cage. Then call animal control and have them come pick up the dog. Do that a few times and when the owner gets tired of paying to get the dog out of animal control, they will either tie him up or get rid of him.
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