Yup. Make each one explain 1) where in the Constitution that program falls under the purview of the Federal Government and 2) why a coal miner, teacher, and single mother should have to pay for them.

Start from zero, and have each program justify its existence ... from an Objectivist viewpoint. I doubt many will make the cut.

(1) would be wholly within Trump's purview and is Constitutionally provided for.

(2) I would note that this is an Executive Branch advisory note written by a Justice Department lawyer. It does not hold the rule of law as such. I would submit that the President would be better off to employ #1 above or #4 below. The duties of the President are to "faithfully carry out and execute the laws of the United States" whether he agrees with them or not. His Constitutional refusal lies solely in the Veto power - if he chooses not to use it or is overridden by Congress, that is where his power to object ceases. Any other form of objection should subject the President to Impeachment proceedings for dereliction of duty and/or obstruction of justice.

Case in point was President Obama's illegal granting of amnesty. This was specifically beyond his powers and for this action not only did the Supreme Court rule against him, but Congress had every right to (and should have IMO) Impeached the President. A President who decides at his own will and pleasure which laws duly passed by Congress to enforce sets himself up as an imperial tyrant and should not be tolerated.

(3) This one is just #2 lite. The purpose is still to take it upon himself to oppose a duly passed and legal law. It is not up to the President to decide what is Constitutional. That role is left for the Supreme Court. He can and should advise Congress (using the Department of Justice) as to the legal and practical viability of various laws, but once the law has been declared legal and has passed through the requisite Constitutional processes, he is under solemn obligation to carry them out.

I would propose that one that is legal is #4 - executive discretion. There is a judgement call involved in carrying out the will of the Legislature as per any Act of Congress. (For a humorous definition of an "Act of Congress", see Dave Barry's book Dave Barry Slept Here.) A President could - under executive discretion - act within the confines of the directive to a limited manner or extent based on the resources granted. This is only a marginal effort, however, and is likely to be a result of partisanship rather than real Constitutional objection.

I see several ways that Trump could defund certain programs even if Congress authorizes them:
(1) Veto any spending bill that authorizes and funds such programs.
(2) Sign such bills ("to avoid a government shutdown") but refuse to spend any money on objectionable programs on the grounds he considers them unconstitutional. (I'm thinking specifically about Planned Parenthood and public broadcasting subsidies.) He does have this authority - see https://fas.org/irp/agency/doj/olc110...
(3) Issue regulations or executive orders to oversee such spending in minute detail and, to the extent possible, slow it down to reduce its effectiveness.

These are just three potential strategies to derail unneeded spending. Undoubtedly there are others.

According to the Constitution, ALL funding bills must originate in the House of Representatives. It's a mere formality/courtesy that the President submits a budget to Congress. As head of the Executive branch, the President is charged with executing the duties laid upon him by the Legislature, so it's quite appropriate in my opinion that he be asked for how much money he thinks should be allocated toward accomplishing those missions. But that doesn't mean that the Legislature has to do what he says. President Obama's budgets (when he even bothered to create them or when Congress actually presented a budget) were frequently so ridiculous not even his own party voted for them (the two budgets he presented tallied a combined ONE vote).

So ultimately, it isn't Trump choosing the eliminate these programs, it would be Congress - by choosing not to fund them. Which seems entirely appropriate given that they had to sponsor them in the first place.

It does seem a little ass backwards to me as well. He can suggest who not to fund but that should be about it.

The article says Congress would have to approve the cuts. I'm not sure why. I thought Congress would have to pass legislation authorizing the spending for the next fiscal year, and that Trump could veto such legislation. How much power does Trump actually have to eliminate programs?