The Americans, working with Iraqi forces in a new drive to reclaim parts of the Iraqi capital from gunmen and bombers, call it Operation Together Forward.
It is key to establishing the authority of Iraq's still relatively new government, to the coalition's handover of full responsibility for security to the Iraqis and - more importantly - to averting what even US officers now acknowledge is the risk of outright civil war.
They are claiming some successes in reducing the number of violent incidents in areas they have already tackled and detaining people alleged to be involved in the violence.
But nowadays who are they actually up against?
The most visible trend in recent months in Baghdad and certain other cities and towns has been the increasing sectarianism - deadly, tit-for-tat violence perpetrated by certain Shia Muslim groups against Sunnis and certain Sunni groups against Shias.
This has happened since an important Shia shrine in the northern town of Samarra was attacked in February.
MAIN ARMED GROUPS IN IRAQ
Some of this sectarian violence is carried out by militias with links to politicians.
But analysts also say there is a blurring between violence that appears to be sectarian or ethnic in its motivation, criminality and also a crude settling of scores.
The sectarian violence has come to overshadow all other kinds.
Meanwhile the Sunni-led insurgency that erupted after the ousting of Saddam Hussein continues despite a reconciliation initiative launched by Prime Minister Nouri Maliki.
It is this entire spectrum of violence that the Americans and Iraqis are attempting to reduce by putting some 12,000 additional security forces into Baghdad between them and conducting sweeps in one volatile district after another.
They have declared themselves to be hunting "death squads" that have left many residents of the city living in fear of a knock on the door or of being gunned down in the street.
The latest phase of the operation quickly provoked political differences between the Americans and the prime minister over a raid in Sadr City, stronghold of the radical Shia cleric Moqtadr Sadr, whose support is important to the ruling alliance.
It showed how easily the strategy can be compromised politically.
Looking after their own
But even if Operation Together Forward were to succeed in its aim of bringing a measurable improvement in people's lives by next month and reversing the flight from mixed areas - a big "if" - the sheer number of groups on the streets who claim to be representing someone's security interests is the underlying problem.
And calls to disband - even more to disarm - the politically-aligned militias are certainly easier said than done.
Now there is a new controversy.
Hadi al-Amiri, an MP and head of a Shia militia, has publicly called for the setting up of people's committees to provide local security - Shias protecting their neighbourhoods, Sunnis their own and joint patrols in mixed areas.
It is an idea that has been touted before - and criticised as an attempt to get around the disbanding of militias.
For the ordinary residents of Baghdad - many with a gun of their own as their last line of defence - it is the desperate wish for better security that probably matters even more than the source of their insecurity.
For the prime minister, time is not on his side if he is to prove that he can deliver genuine security.
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