Why Greed is Often Good: Approximate and Local Submodularity.

The greedy algorithm is perhaps the simplest heuristic in combinatorial optimization and is often used to solve difficult optimization problems. When maximizing a nonnegative, increasing, submodular function, the greedy algorithm has a worst-case performance bound proportional to its optimal objective value at a given maximum cardinality. However, many problems have objective functions that are not submodular. We introduce new metrics that quantify a function's proximity to being submodular. Using our proposed metrics, we derive performance bounds for the greedy algorithm applied to any nonnegative, increasing set function. Insights from our derivations allow us to generalize existing bounds. We examine multiple bounds using generalizations of classical optimization problems that do not generally have submodular objective functions. Our new bounds are competitive with, and often improve, those in the machine learning community and apply to a wider variety of problems than previously considered in the literature.