In the tweets this week, people are suspected of stealing from a Cub Liquor, a Speedway, a nail salon and an auto parts store.
Police say the man was also seen on video throwing objects at police officers.
More than 130 were arrested over last week's rioting.
The owner of Brit's Pub, one of four businesses set on fire, expects months-long recovery
Calls for peace have followed the night of damage and violence.
The department shared the video to "dispel rumors," and deleted the video out of respect for the man and his family.
The Minnesota State Patrol has also been sent to downtown Minneapolis.
The building was targeted while protesters demonstrated outside the nearby 5th Police Precinct.
The damage is estimated at over $500 million.
Walz spoke Friday on WCCO Radio.
The man is accused of helping his brother and sister-in-law avoid arrest.
Charges were filed against a 19-year-old from Maplewood, and a 31-year-old Minneapolis man.
The 22-year-old was captured Sunday in Breckenridge, Colorado.
The Granite City has had consecutive nights of large, unruly crowds after a police incident.
Max It Pawn on E. Lake St. was burned down during the May 28 riots.
The retail chain says it's been making progress in its efforts to reopen.
Rewards totaling $10,000 are being offered to find them.