KABUL, Afghanistan -- International and Afghan troops have begun a key combat phase against insurgents in southern Afghanistan and expect heavy fighting, officials said Monday, in an operation that is crucial to the U.S. strategy to turn around the nine-year war against the Taliban.

The allied forces were moving into two or three areas around Kandahar city in southern Afghanistan at once to pressure the Taliban "so they don't get the chance to run away," Shah Mohammad Ahmadi, chief of Arghandab district northwest of the city, said Monday.

"Before, when we have tried to get rid of the Taliban, when we cleaned one area we found more Taliban in a different one," he said.

A top NATO officer said Sunday that the alliance a few days ago had launched its "kinetic," or combat, phase of "Operation Dragon Strike," a joint military push with Afghan forces around Kandahar intended to rid the area of insurgents and interrupt their ability to move freely and stage attacks.

"It is a significant ground operation with air support," German Brig. Gen. Josef Blotz, a NATO spokesman, said at a news conference. "We expect heavy fighting."

"Afghanistan and coalition forces are destroying Taliban positions so they will have nowhere to hide," Blotz said. "Once this is done, insurgents will be forced to leave the area or fight and be killed."

NATO said militants have fought back with rocket-propelled grenades and small-arms fire.

It said no Afghan or coalition troops have been killed in the operation.

The push in Kandahar is seen as key to the Obama administration's strategy to turn around the nine-year war as insurgents undermine the ability of an Afghan government to rule much of the country. Kandahar remains particularly dangerous; seven U.S. troops have been killed in Kandahar this month. Another three have been killed in the south, but no further details have been released.

"First, they are clearing the area of Taliban and then they are searching the area for mines," said Kareem Jan, district chief of Zhari, west of Kandahar.

In another volatile section of the nation, British officials said Monday that they were in contact with Afghan authorities about the disappearance of a British aid worker and three of her Afghan colleagues. The four were ambushed Sunday as they traveled in two vehicles in northeastern Kunar province in eastern Afghanistan. Police fought a gunbattle with the kidnappers near the attack site before the assailants fled, Kunar police chief Khalilullah Zaiyi said.

Tim Waite, a spokesman for the British Embassy in Kabul, said officials were working closely with all relevant local authorities and said the worker's family had been contacted.

Steven O'Connor, communications director for Development Alternatives Inc., a global consulting company based in the Washington, D.C., area, said late Sunday its employees, including a British national, were involved. The company works on projects for the U.S. Agency for International Development in Afghanistan.

Britain's Foreign Office in London said it could "confirm that a British national has been abducted in Afghanistan."

In the east, meanwhile, NATO helicopters based in Afghanistan carried out at least two airstrikes in Pakistan that killed more than 50 militants after the insurgents attacked a small Afghan security outpost near the border, spokesmen said.

NATO justified the strikes based on "the right of self-defense." Pakistan is sensitive about attacks on its territory, but U.S. officials have said they have an agreement that allows aircraft to cross a few miles into Pakistani airspace if they are in hot pursuit of a target.

The first strike took place Saturday after insurgents based in Pakistan attacked an Afghan outpost in Khost province, which is located right across the border from Pakistan's North Waziristan tribal area, said U.S. Capt. Ryan Donald, a spokesman for the NATO-led International Security Assistance Force in Afghanistan.

"The ISAF helicopters did cross into Pakistan territory to engage the insurgents," said Donald. "ISAF maintains the right to self-defense, and that's why they crossed the Pakistan border."

The strike killed 49 militants, said U.S. Maj. Michael Johnson, another ISAF spokesman.

The second attack occurred when helicopters returned to the border area and were attacked by insurgents based in Pakistan, said Donald.

"The helicopters returned to the scene and they received direct small arms fire and, once again operating in self-defense, they engaged the insurgents," said Donald.

The strike killed at least four militants, said Johnson.