Here's Alaska's budget crisis in a nutshell:

Legislators are writing checks faster than money has been flowing into the state's main bank account.

And unless they stop, according to their own budget analysts, next year the cash finally runs out.

There's one fix that many lawmakers, and Gov. Bill Walker, agree could cover most of Alaska's massive deficit: the $63 billion Permanent Fund, a pool of investments that's been earning billions a year.

But for the legislative session that starts Tuesday, an agreement on how to use the fund appears not much closer than in the past two years, which saw gridlocked lawmakers spending months of extra time fighting in special sessions.

Here's a rundown of Alaska's budget problem, the range of possible solutions and the obstacles to getting them passed.

The problem

Alaska lawmakers have been facing down the same big problem in each of the past three years: How to replace billions of dollars in oil revenue that evaporated when prices crashed in 2014, leaving a huge hole in the budget.

Their solution has taken the form of a once-massive state savings account, the Constitutional Budget Reserve, on which lawmakers have depended to cover their multibillion-dollar deficits.

In this year's session, lawmakers are facing the same problem. Again.

But their solution will have to be different, unless projections from the Legislature's own budget analysts prove to be wrong.

The budget reserve, which held $13 billion less than four years ago, now lacks the cash to cover what's expected to be a $2.5 billion deficit next year, those analysts say.

At stake is the array of programs supported by state government, from health care to public schools to snowplowing.

"That's the situation: 'OK guys, what are you going to do? Because you can't do the budget the way you used to do it,' " said David Teal, the Legislature's chief budget analyst. "We've been saying this for three years, that we're going to run out of money in 2019. And we actually hit that pretty darn close."

The Solution

Alaska's spending amid its budget crisis has fallen sharply between 2015 and its 2018 plan, from $6 billion to $4.3 billion in unrestricted general funds.

But the budget shrinkage has slowed sharply. Cuts to the operating and capital budgets this year totaled just 3.5 percent, or about $160 million, as lawmakers failed to agree on programs — many of them already diminished — that could still be reduced.

Since the deficit remains at more than $2 billion, the most politically realistic fix appears to a big new source of cash, not a major reduction in spending.

Typically, when government needs cash, it raises or levies taxes. But Alaska has an alternative: the Permanent Fund, originally seeded, and still augmented, with oil revenue.

Over its 40-year history, much of the fund's spending has gone toward dividends — cash checks — for Alaska residents. But Walker and legislative leaders in both the House and Senate say that now, the fund should also be used to to help pay for government.

Lawmakers have proposed spending nearly $2 billion a year from the Permanent Fund on government while still paying out reduced dividends to residents. That much cash could erase most of the state's deficit.

An income tax like the one supported by Democratic legislators, and a handful of Republicans, could raise another $700 million, wiping out the deficit completely.

The Politics

On paper, fixing Alaska's deficit is at a "junior-high" degree of difficulty, said Randy Hoffbeck, Walker's former revenue commissioner.

"The math is really pretty easy," Hoffbeck said. But, he added: "The political ideology becomes a real big issue."

That's why the question of just how to tap the Permanent Fund, and whether the state should also start taxing its residents, has stymied legislators for the past two years.

And this year's debate isn't expected to be easier.

A Republican-dominated majority runs the state Senate, and its leader, Senate President Pete Kelly, R-Fairbanks, strongly opposes taxes. Kelly and his colleagues argue that the state's deficit can be reduced to a few hundred million dollars by relying solely on Permanent Fund earnings and spending cuts.

A largely Democratic majority, meanwhile, controls the state House. Its leaders have refused to vote for a restructuring of the Permanent Fund — and the resulting reductions to Alaskans' dividends — unless an income tax also comes with it.

That's because dividend reductions affect each resident equally, regardless of how much money they make — meaning that poor Alaskans would be hit harder, proportionally.

House majority members say an income tax that asks for more from higher-earners would make for a fairer deficit-reduction plan. It would also help insulate the state's finances, they argue, if there's another crash in oil revenue or a stock market correction that reduces the Permanent Fund's value.

Those deep policy disagreements would be difficult to bridge even if legislative leaders enjoyed a good working relationship.

But they don't. When lawmakers last left Juneau, after a special session on crime in November, they issued a stream of statements and attacks blaming each other for passing a bill that attorneys said had obvious constitutional problems.

Since then, things seem to have improved little. At a speech earlier this month, Kelly, the Senate president, said his response to tax proponents would be "mocking laughter."

"I don't think there's probably a lot of trust between either group," said Rep. Mike Chenault, R-Nikiski, the former House speaker. "I don't know how they fix that — maybe it's distrust — between the two. But they're going to have to sit down together and have those conversations and see if there's any items that they can agree on."