The Iraqi government has proven incapable of providing any semblance of law and order quite probably because their democratically elected leaders cannot or will not make the tough decisions necessary to stop the violence and find a "political solution".
I'm not a counter-insurgency or civil war expert by any means, but giving American forces a freer hand in battling militias is certainly one viable approach to slowing the carnage. But as long as we continue to respect the sovereignty and prerogatives of the internationally recognized government in Baghdad, our hands are tied. Increasing, maintaining, or drawing down our troop level in this context is immaterial to the outcome because they can't act decisively while leaving our soldiers without a sense of mission.
So, why don't we ask the Iraqi people a simple question in a free and fair plebiscite.
Do you want American troops to remain in Iraq, and if so are you willing to confer upon them the legitimacy to take whatever actions they deem necessary to restore law and order? The authority to impose martial law as needed would be renewable every 18 months.
This wouldn't be like a referendum in the states where hyperbole and innuendo rule the day. The information campaign describing what American generals truly believe is necessary to restore law and order would be sober and presented dispassionately.
Moreover, a plebiscite might give Iraqi leaders the political cover (and American protection detail) that allows them to make some really tough and necessary choices, or alternatively provide US with the mandate to come home.