Being a dog owner

I recently learned something about being a dog owner. The dog is charming, sweet, loyal, handsome. A positively protective and willing to please animal. Until the neighbors cat gets involved. There are hundreds of cats roaming around the area. There is one cat in particular that haunts our back yard. This cat taunts our dog. She sits in high places and meows and hisses at the dog. She jumps down from her perch and goads the dog into chasing her to places the dog can't go. Our dog can be heard, from time to time, trying to flush this cat out from under the deck. The owners of the cat are quite aware that this happens.
The cat also uses our yard for a toilet. She digs in my garden and eliminates there, covering her 'stuff' with my mulch, displacing bulbs, tearing at roots, and killing my plants. She sleeps on our porch and prances around our yard as if it is hers. She hisses and scratchs and nips at my children if they try to pet her - while she is in our yard.
On occassion, my small children open the front door while the dog is in the house. She sees this as the perfect occassion to make a run for it. She knocks the small child over to get out of the front door. Once out there, she doesn't listen to anyone for any reason. No food can tempt her to return to her back yard. Not the promise of a walk. No sweet cajoling with offers of petting or belly scratching. Nothing.
Enter stage right, the cat. The cat is on top of a car, sees the dog, and carrys on her tradition of teasing. The dog, no longer confined to a fenced area, takes off after the cat. The cat jumps off of the car and darts underneath it. The dog ensues, barking, growling and snarling at this cat. The neighbor hears and sees it. Screams at me to get my *nasty words* dog off of his cat.
The dog, for at least 3 years, has yet to catch this cat in it's own yard. The barking, growling, snarling and chasing of the cat by my dog, in the dog's confined space was never a problem. Now, out in the open, it's MY problem. Really?
I make more of an effort to keep the dog inside. The children know to put the dog in the back yard before they open the front door. The dog is very rarely out in the neighborhood.
The cat? She still wanders at will. She still shows up in the yard, teasing the dog. The neighbor still doesn't care that the cat defiles my gardens and riles the dog. My dog still can't catch the cat.