A group of businesses may agree to a set of conventional standards in the form of a charter. This charter would/could be adopted by any number of arbitrators in order to streamline dispute resolution. This is only one possible solution.

One of the more important elements of security in a market driven society is insurance and reputation, as opposed to self defense or militarization alone. Anything that cannot be financially tolerated through the purchase of insurance would in general be something that may be difficult to carry out. I'm anticipating the "mad scientist" scenario here by outlining the difficulty of being a 'bad neighbor' and introducing a kind of risky behavior that could have potential harm to neighbors.

As for your example, there is an incentive to be part of a dispute resolution system, so you may be volunteering for all the merits of protection as well as being subject to arrest. Of course some people might not want to be party to any such contract, but it would be to their detriment if they were also prone to violence since it may be more difficult to gain sympathy if they had no form of representation given a society in which it was a voluntary choice to find one.

Detaining people for violating the property or person of someone else is a much more clear-cut thing when a person is caught in the act or video is being recorded. The question of arrest and legitimacy after a period of time has passed or there are doubts about the legitimacy of apprehending a suspect becomes a more complex issue. In terms of wrongful arrest, I think a voluntary society would at least be more compensatory for such actions. As it is now the state doesn't really apologize for taking up your time or money to fight charges, so I believe you could have more justice without a monopoly on force trying to process people through a system.

I'll just finish by saying that the Non Aggression Principle is not a cure-all. It can't give us all the answers for how to deal with all the possible situations we may find ourselves in. We also have to apply our minds to solving these problems and discovering better market solutions, so the NAP and market action are two sides of the same coin.