Committing a local choice
An optimization problem can admit many feasible solutions. A greedy method selects a locally preferred candidate, removes incompatible alternatives, and repeats on what remains instead of fully exploring every combination.
The local rule must be explicit. Earliest start, earliest finish and shortest duration are different interval-selection rules. Their intuitive appeal does not make them interchangeable; correctness is tied to a particular rule, objective and constraint set.
References: [1]
An exchange argument for interval scheduling
For nonoverlapping intervals on one resource, maximizing the number of selected requests permits choosing the available earliest-finishing interval repeatedly. An earliest-starting long interval can instead block many later requests.
An exchange argument justifies the choice: replace the first interval of an optimal schedule with the greedy one, which ends no later. The remaining intervals still fit and cardinality is unchanged. Applying the argument to the remaining problem connects each local step to a complete optimum.
References: [1]
Changed objectives need new proofs
With unequal rewards and a maximum-total-value objective, earliest finish is no longer generally optimal. A short low-value request can exclude a much more valuable overlapping one, and the cardinality exchange proof does not preserve reward.
Minimum-spanning-tree greedy rules have a cut-property justification; weighted interval scheduling can use dynamic programming. For an unproved greedy heuristic, practical convenience should be distinguished from an optimality guarantee.