Fighting raged across southern Afghanistan on Sunday as the U.S. military pounded suspected Taliban positions with airstrikes that killed at least 15 fighters along a narrow mountain footpath.

A Taliban spokesman, meanwhile, asserted that his fighters had assassinated a kidnapped Afghan police chief and five of his men for collaborating with U.S.-led forces.

U.S. aircraft opened fire on a group of suspected Taliban fighters in the high mountains northwest of Gereshk, in southern Helmand province, after rebels had pinned down a coalition ground patrol with fire from rockets and small arms.

"Initial battle-damage assessments indicate 15 to 20 enemies died and an enemy vehicle was destroyed," the Army said in a statement. No Americans were injured.

Military spokesman Lt. Col. Jerry O'Hara added a warning to the insurgents.

"When these criminals engage coalition forces, they do so at considerable risk," he said. "We are not going to let up on them. There is not going to be a safe haven in Afghanistan."

O'Hara said additional U.S. and Afghan forces had been sent to the scene and that the numbers of rebel dead could rise.

Three months of bloodshed across the south and east has left hundreds dead and sparked fears that the Afghan war is widening, rather than winding down.

Afghan and U.S. officials have warned that they expect more bloodshed ahead of parliamentary elections in September.

In other violence, three rockets smashed into the southern city of Kandahar, jolting residents but causing no casualties.

One rocket hit an empty lot near the former home of fugitive Taliban leader Mohammad Omar, which now houses U.S. Special Forces troops, said Gen. Salim Khan, the deputy police chief.

On Saturday night in Helmand, gunmen shot to death three men -- a judge, an intelligence worker and an employee of the provincial education department, said Haji Mohammed Wali, a spokesman for the governor. He said it was not clear whether the Taliban or some other armed group was behind the attack.

About 260 suspected rebels and 29 U.S. troops have been killed since March, according to Afghan and U.S. officials. About three dozen Afghan police officers and soldiers also have been killed, as have more than 100 civilians.