Congress authorizes appropriations for new expenses (so that executive branch could not authorize their own expenses at taxpayer cost), but not cost cutting within executive branch
Article I, Section 9 grants Congress the power of the purse to approve spending and article II Section 3 says that the president must "take care that the laws be faithfully executed". That's all of the authority over the budget the president has: the ability to faithfully execute the spending Congress approved or to veto the bill when it's passed.
Further, the matter has been to the supreme court TWICE already. First during Nixon's reign in the 70's (Nixon lost), then later under Clinton during the line-item veto mess. Clinton also lost, rightfully.
The worst part of all of this is that I know these things and I'm NOT someone who swore to uphold the constitution.
Even if you agree with the goals, the methods are not just bad... they're illegal, ineffective, and likely to be very expensive in the long run. They should have just done it properly.