Best Algorithms for Approximating the Maximum of a Submodular Set Function

A real-valued function z whose domain is all of the subsets of N = {1,..., n is said to be submodular if zS + zT ≥ zS ∪ T + zS ∩ T, ∀S, T ⊆ N, and nondecreasing if zS ≤ zT, ∀S ⊂ T ⊆ N. We consider the problem maxS⊂N {zS: |S| ≤ K, z submodular and nondecreasing, zO = 0}. Many combinatorial optimization problems can be posed in this framework. For example, a well-known location problem and the maximization of certain boolean polynomials are in this class. We present a family of algorithms that involve the partial enumeration of all sets of cardinality q and then a greedy selection of the remaining elements, q = 0,..., K-1. For fixed K, the qth member of this family requires Onq+1 computations and is guaranteed to achieve at least $$\biggl[1-\biggl\frac{K-q}{K}\biggr\biggl\frac{K-q-1}{K-q}\biggr^{K-q}\biggr]\times100 \quad\mbox {percent of the optimum value}.$$ Our main result is that this is the best performance guarantee that can be obtained by any algorithm whose number of computations does not exceed Onq+1.