Generalized Skyline Interval Coloring and Dynamic Geometric Bin Packing Problems

We consider two combinatorial optimization problems, named Generalized Skyline Interval Coloring (GSIC) and Dynamic Geometric Bin Packing (DGBP). The input to both problems is a set of interval jobs, with each job specified by a horizontal active time interval and a vertical size. For GSIC, each job is to be allocated a vertical interval of the specified size in the range [0, +∞). For any two jobs with overlapping active intervals, their vertical intervals must not overlap. The instantaneous cost of an allocation at any time is defined as the highest point allocated to the active jobs. The target is to minimize the accumulated cost over time. For DGBP, each job is to be assigned to a machine of capacity g and be allocated a vertical interval of the specified size in the range [0, g). For any two jobs with overlapping active intervals, if they are assigned to the same machine, their vertical intervals must not overlap. The target is to minimize the total machine busy time, where the busy time of a machine is the time duration in which there is at least one active job in the machine. We develop O(1)-approximation algorithms for both problems in the offline setting and asymptotically optimal algorithms in the non-clairvoyant and clairvoyant online settings.