Greedy Switching Functions on Time Scales

Submodular functions are an important class of real-valued functions whose domains are the power set of some finite collection of items. An existing result states that, given an monotonic submodular objective function, a greedy algorithm achieves an approximation ratio of at worst 1-1/e. More recently, a similar result was shown for functions whose domain is a class of objects called continuous sequences. We generalize both of existing works by defining a class of objects called switching functions. Suppose that there is a system with a finite number of possible configurations and a set of times at which the configuration may be altered; this set of times may be discrete or continuous. A switching function represents a policy that chooses a configuration at each point in time, and that switches the configuration a finite number of times. We show that, given some regularity conditions reminiscent of submodularity, a greedy algorithm for constructing switching function provides an approximation ratio. This generalizes existing results concerning submodular functions and sequence submodular functions. This provides a unified approach to greedy algorithms with qualities analogous to submodularity in discrete and continuous settings.