The US government has been deeply concerned - and some may say overly so - with the spread of terrorism in the region. This week we are reminded that among what was perceived by many as a largely flawed policy causing many problems, few of them are as severe as those caused by Washington's approach to the Yemen situation.

The corrupt, and largely ineffective regime of President Ali Abdullah Saleh has weathered months of mass protests largely because of unwavering, if soft-spoken, support from the US (and from neighbouring Saudi Arabia). What the US gets from Mr Saleh is a free hand to monitor, hunt and kill members of Al Qaeda in the Arabian Peninsula (AQAP).

Fatally short-sighted, US policy is unable to look beyond the "war on terror". If it could it would by now have pensioned off Mr Saleh and encouraged compromise among other factions in Yemen.

But while Mr Saleh remains in Riyadh, recovering from an assassination attempt in June, Yemen's security has been deteriorating sharply. This result is counter to broad American security goals and disastrous for Yemen's 24 million people. Poor and badly run for decades, Yemen is now tipping into failed-state status.

Sanaa, the capital, is seeing a three-cornered struggle among elites: Mr Saleh and his family (his son, Ahmed, commands the powerful Republican Guard); the rival Ahmar family; and the faction of Ali Mohsin, a general who broke with the regime. Meanwhile ordinary Yemenis protest to demand a country without Mr Saleh.

But while thugs with military weapons, and forces with assorted loyalties, have killed dozens of protesters in recent days, something different is happening elsewhere: "While elites fight over the carcass of the state," says Princeton University Yemen expert Gregory Johnsen, "on the periphery … other actors are busy gobbling up as much territory as they can." In the south these include Islamic militants and some tribes, and in the north the Huthi faction.

As Yemen nears disintegration its economy is in tatters, hunger is increasing and violence growing. Meanwhile, White House security advisor John Brennan says smugly that "counter-terrorism cooperation with Yemen is better than it's been during my whole tenure".

He will not, we fear, be able to say the same next year if AQAP and other extremists are operating freely in a collapsed state. When, if ever, will the US learn that it cannot make itself safer by backing dictators, resisting justice and stifling the aspirations of long-suffering peoples? Short-sighted security goals may play well with American voters, but they get Yemenis nowhere.