At least six insurgents and five Russian troops were killed in gun battles in the mountains of Chechnya, while five people were killed in attacks in Makhachkala, capital of neighbouring Dagestan, officials said.
Near-daily attacks in Chechnya, Ingushetia and Dagestan, mostly targeting law enforcement and government officials, have alarmed the Kremlin in recent months.
President Dmitry Medvedev has called the violence Russia's biggest domestic political problem and last month appointed a businessman as his envoy to the region to tackle underlying causes such as unemployment and corruption.
City police chief Akhmed Magomedov was killed along with his driver and two bodyguards when gunmen opened fire on his car in Makhachkala. He died on the way to hospital, said police spokesman Mark Tolchinsky.
The head of a police counterterrorism department in one of Dagestan's districts was killed earlier in the day when a bomb planted beneath his car exploded, the federal Investigative Committee said.
In neighbouring Chechnya, the scene of two devastating 1990s wars, forces controlled by Moscow-backed Chechen President Ramzan Kadyrov battled rebels in the forested Caucasus Mountain foothills southwest of the capital Grozny.
Kadyrov has expressed confidence the insurgents will soon be destroyed and has even announced plans to build a ski resort once they are eliminated from strongholds in the southern mountains. He launched a campaign to root them out last month.
Five federal servicemen were killed in fighting that began on Thursday and persisted on Friday said Maryam Nalayeva, an official in Chechnya's Investigative Committee. Six insurgents were killed on Thursday in fighting nearby, Kadyrov's office said.
Russian news agencies cited investigators in Chechnya as saying fighting initially broke out in one location and then erupted again in a neighbouring district after government forces tracked down rebels following the first gun battle.
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