I don't know about New York laws but I assume they would be similar to ours in Minnesota. Minnesota laws are very clear about killer or nuisance dogs & cats. Following read:
347.03. Dogs may be killed
Any owner or caretaker may kill any dog found chasing, injuring, or worrying sheep or other livestock or poultry owned by or in care of such owner or caretaker, on lands or premises owned or controlled by the owner or caretaker, and any owner or caretaker of sheep may kill any dog found on the owner's or caretaker's premises where sheep are kept, not under human restraint or control.
Cats are not mentioned in the above paragraph but the heading for this section of Minnesota law reads: Police Regulations. Chapter 347. Dogs and Cats.
Just do a google search using the key words - New York dog laws - and you should find exactly how your laws read.
Saying all this I would try a little sugar first. After all you catch more flies with sugar. My closest neighbor has two big dogs and they raided my yard once last summer. Luckily I was home and intercepted them before they did any damage. I chased them home with a big stick and later had a talk with the owners. They have been very good about keeping their dogs under control and I let them know how much I appreciated that. They also appreciated my actions, apologized and promised to keep the dogs under control. its been almost a year and there have been no further troubles. I hesitate to start killing right away because even tho we love our chickens we also have two dogs and also love them. There is so much violence and killing in the world I think its worth the effort to at least try and talk things over first. Good luck