Constraint Satisfaction: maximizing total value with no overlaps
You are looking for a maximum weight independent set in an interval graph, which can be solved in linear time (by a deterministic algorithm). By the way, the same is true also for a superclass of interval graphs, namely chordal graphs.

One such algorithm is based on dynamic programming, you can see e.g., [1] for details and discussion.



[1] Hsiao, Ju Yuan, Chuan Yi Tang, and Ruay Shiung Chang. "An efficient algorithm for finding a maximum weight 2-independent set on interval graphs." Information Processing Letters 43.5 (1992): 229-235.