GROZNY, Russia — Russian troops and Chechen rebels struggled Monday for control over several Grozny neighborhoods, while militants attacked federal forces in two regions south of the capital.

Skirmishes flared all along Grozny's perimeter, the Interfax news agency reported. The heaviest fighting was reported in the district around Minutka Square in central Grozny, which rebels control. The square leads to a strategic bridge over the Sunzha River, which the Russians seized last week.

Russian troops sent salvo after salvo of shells screaming into Chechen-held parts of the capital, while troops fanned out across Russian-controlled neighborhoods to check for hidden snipers and arms caches.

Federal troops backed by three Mi-26 helicopters drove into a central district of Grozny on Sunday and dislodged rebels from a five-story building they had been using as a hideout, an officer who took part in the fighting said.

But the Russians, targeted by snipers in nearby buildings, lost at least five men in the battle.

In some areas, Russian troops have been advancing building by building during the day only to retreat at night, when the rebels tend to stage ambushes.

With temperatures falling below minus 14, cold weather has also been taking a toll on Russian troops. Over the past two days, 35 soldiers from the village of Staraya Sunzha, just outside Grozny, have been hospitalized with frostbite.

And soldiers have complained of poor coordination between the Defense Ministry and Interior Ministry troops. Sgt. Grigory Merenkov, an Interior Ministry officer, said his seven-man unit was caught in a rebel ambush on Sunday night and asked for mortar support from the army to cover its escape. No artillery barrage followed.

"We asked again after 15 minutes. A voice asked if we really needed the support. And in five minutes, they pounded the ditch we were hiding in," Merenkov recounted. A shell tore through one of his colleague's legs, he said.

Still, the military has continued to insist that its weeklong drive to reach the city center was making progress.

It said that five Russian soldiers had been killed and 11 wounded on Sunday, while close to 80 rebels were killed, the ITAR-Tass news agency reported. The claim could not be confirmed, and each side tends to underestimate its own casualties and overstate the enemy's.

Interfax reported that more than 30 Russian troops had been killed and about 70 wounded over the weekend.

The disparity in reports reflected growing public questioning of the official casualty count. The respected Soldiers Mothers' Committee said today that based on reports from its regional members, it estimated 3,000 servicemen had been killed, compared with the approximately 800 the government has reported.

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"We are inclined to believe what we find ourselves," Valentina Melnikova, spokeswoman for the committee, told Echo of Moscow radio. "We don't know what methods the military uses. It's on their conscience."

Rebels attacked federal forces in the south on Sunday, trying to break through from the rebel stronghold of Vedeno southwest to Shatoi in the rugged mountains near the Georgian border, said Lt. Col. Alexander Sapronov, a military spokesman. He said that the Russians had rebuffed the assault.

In Moscow, Russian First Deputy Prime Minister Mikhail Kasyanov said the government had spent about $178.5 million on the Chechnya campaign, exceeding the planned spending by about $53.5 million.

Russian troops entered the breakaway republic of Chechnya in September after Chechen-based militants stormed villages in the neighboring Russian region of Dagestan. The rebels also were blamed for a series of deadly bombings of apartment houses in Moscow and two other cities in September.

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