Rich in unexploited natural resources, Greenland gained autonomy from Denmark in 1979 and was granted self-rule in 2009, though Copenhagen retains control of foreign and defence affairs.
The vast island between the North Atlantic and Arctic oceans, mostly covered in ice and home to just 55,000 people, also receives some 3.6 billion kroner (483 million euros, $591 million) in subsidies each year from its
former colonial master.
While Denmark's constitution recognises the island's right to decide on its own independence, if it became a sovereign nation it would lose the much-needed subsidies, which make up 60 percent of Greenland's annual budget.
So the main issue for pro-independence campaigners is when to secede, and how to do so without impoverishing the island.
With a gross domestic product of $2.2 billion in 2015, an independent Greenland would be the poorest country in Europe along with San Marino.
Of the seven political parties, six are in favour of independence. Some are keen to declare independence by 2021 to coincide with the 300th anniversary of Denmark's occupation, though most have not set a timeline.
Opinion polls suggest the left-green Inuit Ataqatigiit (IA) party will win Tuesday's election, where 31 seats are up for grabs.
A poll published on April 20 credited IA with 31 percent of voter support, ahead of its main rival, the social democratic Siumut party which has dominated Greenland politics since 1979 and is currently in power.
Seen garnering 27.4 percent of votes, Siumut could find itself relegated to the opposition -- though one in four voters is still undecided.
The two parties are at odds over the use of the island's lucrative natural resources and the thorny issue of uranium mining, which IA, with strong support among urban youths, opposes.
Meanwhile, polls show the newly-formed Cooperation Party, the only anti-independence party, with around 2.9 percent of votes.
Aaja Chemnitz Larsen, a lawmaker for IA, told AFP that before setting a timeline for independence, the island should first lay the financial groundwork.
"Foreign investments are going to be crucial when you talk about the development of Greenlandic society," she said.
Her party wants to see a diversification of investments, as rising temperatures in the Arctic melt Greenland's ice sheet, exposing mineral riches -- and drawing eager glances from the West, Russia and China.
While not sustainable, "economic development the last (few) years has been rather good; the fishing industry has been doing quite well ... Employment has been increasing and unemployment is low," said Torben Andersen, Aarhus University economics professor and chairman of the Greenland Economic Council.
The fishing industry, which accounts for 90 percent of Greenland's exports, is temporarily benefiting from climate change as the rising temperatures bring new species to fish, though that is likely to change over time.
And while Greenland may have a wealth of untapped natural resources, primarily minerals, that could help finance its independence, "it suffers from a lack of infrastructure and a qualified labour shortage," stressed Mikaa
Mered, an Arctic expert and economics and geopolitics professor at France's School of International Relations.
That's also a major hurdle for the territory's desire to emulate the thriving tourism industry of its Arctic neighbour Iceland.
Voter turnout is typically high in Greenland, around 70 percent. Heidi Moller Isaksen, a 51-year-old secretary who lives in the capital Nuuk, declined to disclose which party she would vote for. But breaking free from Denmark is a long-term goal.
"I do want independence one day but we've got to be realistic and take one step at a time," she told AFP. "We can never have independence as long as we have so many social problems."
The Inuit like other indigenous populations are torn between tradition and modernisation. The tension has led to Greenland having one of the world's highest suicide rates, and a third of children are victims of sexual abuse.
In addition, global warming has escalated the exodus from isolated villages to the few towns and cities, said Mered.
It is "wreaking havoc on Greenland's culture: young people are losing interest in traditional hunting and fishing, it's difficult to travel by dogsled from one village to another, and wild animals are moving further and further away from the regular hunting grounds," he said.
All of this leads to "numerous new problems, such as youth suicides."