I’d like to suggest the following for the above description:
This is the Branch and Bound Coin Selection algorithm designed by Murch. It searches for an input set that can pay for the spending target and does not exceed the spending target by more than the cost of creating and spending a change output. The algorithm uses a depth-first search on a binary tree. In the binary tree, each node corresponds to the inclusion or the omission of a UTXO. UTXOs are sorted by their effective values and the trees is explored deterministically per the inclusion branch first. At each node, the algorithm checks whether the selection is within the target range. While the selection has not reached the target range, more UTXOs are included. When a selection’s value exceeds the target range, the complete subtree deriving from this selection can be omitted. At that point, the last included UTXO is deselected and the corresponding omission branch explored instead. The search ends after the complete tree has been searched or after a limited number of tries.
The search continues to search for better solutions after one solution has been found. The best solution is chosen by minimizing the waste metric. The waste metric is defined as the cost to spend the current inputs at the given fee rate minus the long term expected cost to spend the inputs, plus the amount the selection exceeds the spending target:
waste = selectionTotal - target + inputs × (currentFeeRate - longTermFeeRate)
The algorithm uses two additional optimizations. A lookahead keeps track of the total value of the unexplored UTXOs. A subtree is not explored if the lookahead indicates that the target range cannot be reached. Further, it is unnecessary to test equivalent combinations. This allows us to skip testing the inclusion of UTXOs that match the effective value and waste of an omitted predecessor.
The Branch and Bound algorithm is described in detail in Murch’s Master Thesis: https://murch.one/wp-content/uploads/2016/11/erhardt2016coinselection.pdf
Edit: Cherrypicked first and third of instagibbs changes from below. ;)