Set Cover with Requirements and Costs Evolving over Time

Abstract

We model certain issues of future planning by introducing time parameters to the set cover problem. For example, this model captures the scenario of optimization under projections of increasing covering demand and decreasing set cost. We obtain an efficient approximation algorithm with performance guarantee independent of time, thus achieving planning for the future with the same accuracy as optimizing in the standard static model.

From a technical point of view, the difficulty in scheduling the evolution of a (set cover) solution that is “good over time” is in quantifying the intuition that “a solution which is suboptimal for time t may be chosen, if this solution reduces substantially the additional cost required to obtain a solution for t′ > t′′. We use the greedy set picking approach, however, we introduce a new criterion for evaluating the potential benefit of sets that addresses precisely the above difficulty.

The above extension of the set cover problem arose in a toolkit for automated design and architecture evolution of high speed networks. Further optimization problems that arise in the same context include survivable network design, facility location with demands and natural extensions of these problems under projections of increasing demands and decreasing costs; obtaining efficient approximation algorithms for the latter questions are interesting open problems.