More than 20 insurgents including Arab, Chechen and Pakistani fighters have been killed by NATO and Afghan forces who are ramping up operations in the east against a Taliban faction linked to Al Qaeda, the international coalition said Saturday.

Separately, three more NATO troops — an American, a Briton and an Australian — were killed in separate insurgent attacks in the volatile south, officials of the three countries said Saturday.

This comes as the new U.S. commander in Afghanistan says the goal of starting an American pullback by July 2011 will depend on conditions at the time.

In an interview taped for NBC's "Meet the Press" this weekend, Gen. David Petraeus hedged a bit when asked if the summer deadline next year was non-negotiable.

Petraeus said President Obama has "expressed very clearly that what he wants from me is my best military advice." The general, who succeeded the ousted Gen. Stanley McChrystal, said that if sufficient progress has not been made by that date, "I would communicate that to him. ... That's real life."

Petraeus also participated in a video teleconference Friday between Obama Afghan President Karzai, in which the leaders agreed that the United States and Afghanistan should continue to work together to keep the pressure on the Taliban, fight terrorism and protect the Afghan people.

The NATO, Afghan joint force operation began Wednesday against dozens of insurgents holed up in a mountainous area of Zadran district of Paktia province. The operation focused on disrupting the Haqqani network's movement in an area used to stage attacks in the capital, Kabul, and along a highway that links Khost province and Gardez, the provincial capital of Paktia, NATO said.

More than 20 insurgents have been killed, the coalition said. Combined security forces also discovered and destroyed multiple explosive devices and bomb-making equipment, including trip wire and blasting caps, weapons and ammunition. A coalition airstrike destroyed an enemy ammunitions bunker, NATO said.

Three small children were killed and their mother was wounded when a civilian house was hit by an insurgent rocket in Khost city late Friday, provincial spokesman Mubarez Zadran said. He said the insurgents appeared to have been aiming at a coalition base but missed.

The U.S. considers the Haqqani group, led by Jalaluddin Haqqani and his son Sirajuddin, as one of the most dangerous Taliban networks because of its links to Al Qaeda. The group is suspected of playing a major role in the Dec. 30 bombing of a CIA base in Khost as well as a series of attacks in Kabul. It is based in the western border area of Pakistan, where U.S. forces cannot operate on the ground.

"The Haqqani network continually seeks to establish strongholds in the Khost-Gardez pass, disrupting the local government and facilitating the movement of foreign fighters, explosives and weapons into Afghanistan," said U.S. Army Col. Rafael Torres, a NATO spokesman.

Two other operations in June resulted in the deaths of more than 50 Haqqani fighters. Afghan and coalition forces killed 17 fighters including a commander, Fazil Subhan, during a fierce firefight in Khost province June 9, NATO said.

A week later, in the Jani Khel district of Paktia province, Afghan and coalition forces killed at least 35 insurgents including several key leaders for both the Haqqani and Taliban networks. The security force killed Hamiddullah, a Haqqani commander for Sabari district in Khost province who had direct ties to Haqqani senior leadership based in Pakistan and was reportedly responsible for an ambush of an Afghan National Army unit in March, which killed three Afghan soldiers.

In the two-day offensive on the largest foreign fighter camp in the area, assault forces also killed Qari Ismail, a Taliban leader for Jani Khel district, and Maulvi Sadiq, who worked to bring foreign fighters into Afghanistan, NATO said.

Elsewhere, two private security guards and two militants were killed in a gunbattle late Friday in the Karukh district of Herat province, police spokesman Noor Khan Nekzad said. He said 12 other guards were wounded in the skirmish.

NATO officials had no further details about the death of the American service member Saturday. But the Australian military said a member of the country's elite Special Air Service Regiment was shot early Saturday while assisting in a "disruption operation" in the northern part of Kandahar province.

The British Ministry of Defense said Saturday that a British soldier serving with a mine-clearing unit died in a firefight the day before in the Nad-e-Ali district of Helmand province.

Four police were killed and four others were wounded when their vehicle hit a roadside bomb Saturday in Gereshk district in Helmand province, provincial spokesman Daud Ahamdi said.

Also in the south, three Afghan civilians were killed and another was wounded by insurgents in three separate incidents in Kandahar province on Friday. Two of the civilians were killed when a rocket-propelled grenade hit their vehicle in Arghandab district. Another was fatally stabbed by insurgents near the governor's compound in Kandahar city.

In the north, NATO and coalition troops killed two insurgents Saturday after a patrol came under small arms and rocket-propelled grenade fire in Kunduz province, NATO said.

The Associated Press contributed to this report.