Three women and a man were treated for the effects of breathing in smoke following the fire in Murrayburn Green, Sighthill, on Sunday night.
It is thought abandoned furniture was deliberately set alight in the building's common close at about 21:00.
Two of the woman were taken to hospital for medical checks. A total of eight homes had been evacuated.
Police said they were keen to speak to anyone who could help them trace those responsible.
Insp David Ferguson of Corstorphine Police Station said: "Thankfully no-one was seriously injured in this incident but it could have been much worse.
"I am keen to hear from anyone who was in the area around 21:00 on Sunday night and who might have information that can assist us with our inquiries.
"I'd also like to remind residents not to leave items of furniture or rubbish in communal areas where they can be a target for fire-raising."

Sidwell made 34 league appearances last season as the Seagulls returned to the top flight after a 34-year absence.
The 34-year-old joined Brighton on an initial loan deal from Stoke in January 2016, before joining the club on a free transfer ahead of last season.
"Steve has been a key member of the squad over the past 18 months," manager Chris Hughton said.
"Our fans know his qualities well, and his experience, particularly at Premier League level, will be important for us."

A total of 28% had not talked about the symptoms, the University of Aberdeen paper revealed.
Lead author Dr Peter Murchie said it highlighted the complications surrounding emergency cancer diagnosis.
Cancer Research UK said it was a complex area.
The study - published in the British Journal of Cancer - said that, of those who had previously flagged symptoms to their doctor, 81% had been appropriately referred on by their GP for further treatment or investigation of their symptoms.
However it said 19% could be viewed as missed opportunities for earlier diagnosis.
The study, funded by the Scottish Chief Scientist's Office, involved a review of the case notes from 1,800 patients diagnosed with cancer in northern Scotland and registered at GP surgeries across NHS Grampian, Orkney and Shetland.
Dr Murchie, a GP and researcher at the University of Aberdeen, said: "This study sheds light on the complications surrounding emergency cancer diagnosis and how quickly circumstances can change for the patient.
"It's vital we do more to ensure people with persistent or unusual symptoms feel confident about getting them checked out by a doctor without delay.
"And it's equally important to give GPs better access to the tests and specialist advice they need to help them spot potential cancer symptoms at the earliest stage."
Jodie Moffat, Cancer Research UK's head of early diagnosis, said: "For some patients, emergency presentation may be difficult to avoid, but for others there are often things that could have been done differently.
"Studies like this help us to understand this complex picture and identify what needs to change.
"Cancer patients diagnosed as an emergency are more likely to have late stage disease and poorer survival. So it's vital we do all we can to break down barriers to people visiting their doctor with symptoms that could be cancer.
"Cancer Research UK is also working with GPs in Scotland to help improve early cancer diagnosis and ensure they have the freedom to refer patients for further tests and access specialist advice if cancer is suspected."
A blaze at a block of flats in an Edinburgh housing estate was started deliberately, police believe.
[NEXT_CONCEPT]
Brighton midfielder Steve Sidwell has signed a new one-year contract with the newly-promoted Premier League club.
[NEXT_CONCEPT]
More than a quarter of cancer patients diagnosed as an emergency in the north of Scotland had not discussed any relevant symptoms with their GP beforehand, according to a new study.