Arsen Avakov said that government troops broke through rebel positions around the village of Semenovka on the eastern fringe of Slovyansk. "An active offensive stage of the counterterrorist operation is underway in Slovyansk," he wrote on his Facebook page.
Local residents said that several Ukrainian combat jets and helicopter gunships attacked rebel positions on the eastern outskirts of Slovyansk, and heavy artillery barrages have continued throughout the day.
An AP journalist heard sustained gun and artillery fire and saw plumes of black smoke rising over the city.
Avakov warned residents in Slovyansk and nearby cities of Kramatorsk and Krasny Liman to stay at home.
A Ukrainian military officer, who spoke on condition of anonymity because he wasn't authorized to speak to the media, said one serviceman was killed and 13 others were wounded when their vehicle came under rebel fire near Slovyansk.
The Interfax news agency quoted Vyacheslav Ponomarev, the leader of insurgents in Slovyansk, as saying that his men downed a Ukrainian jet and a helicopter gunship. His claim couldn't be independently confirmed.
Ukrainian authorities have repeatedly announced an escalation in armed operations, only to eventually back down. Government forces have in recent days been noticeably reinforced to the north of Slovyansk, however, and deployment of air power over the past week has signalled increased determination.
Slovyansk, which sits on a strategic highway, has seen daily fighting between government forces and the rebels, who have seized government buildings and set up checkpoints around the eastern provinces of Donetsk and Luhansk that form Ukraine's industrial heartland.
The fighting has escalated following the May 25 presidential election won by billionaire candy magnate Petro Poroshenko, with rebels launching an attack on Donetsk airport and shooting down a government helicopter over Slovyansk.
On Monday, hundreds of rebels armed with automatic weapons and rocket-propelled grenades mounted a daylong siege of a border guards base on the outskirts of Luhansk that coordinates the protection of Ukraine's border with Russia. Border guards said they killed at least five rebels in repelling the attack.
Ukraine's acting chief prosecutor, Oleh Makhnitskiy, said Tuesday that 181 people, including 59 Ukrainian servicemen, have been killed in fighting in eastern Ukraine and another 293 have been wounded. With the insurgents controlling several cities in the east, the death toll released by Makhnitskyi is likely to be incomplete.
Efrem Lukatsky near Izyum, Ukraine contributed to this report.
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