Solving a variant of interval scheduling problem
It seems that you are looking for the minimum maximal independent set of an interval graph. A minimum maximal independent set in a graph is the same as a minimum weight independent dominating set, with all vertices weights equal to 1. Since interval graphs are a special case of chordal graphs, Martin Farber's paper Independent domination in chordal graphs (Operational Research Letters, 1(4):134&ndash;138, 1982; ACM Digital Library) contains the algorithm you are looking for.