Greedy Set-Cover Algorithms

1 PROBLEM DEFINITION Given a collection S of sets over a universe U , a set cover C ⊆ S is a subcollection of the sets whose union is U. The set-cover problem is, given S, to find a minimum-cardinality set cover. In the weighted set-cover problem, for each set s ∈ S a weight w s ≥ 0 is also specified, and the goal is to find a set cover C of minimum total weight s∈C w s. Weighted set cover is a special case of minimizing a linear function subject to a submodular constraint, defined as follows. Given a collection S of objects, for each object s a non-negative weight w s , and a non-decreasing submodular function f : 2 S → R, the goal is to find a subcollection C ⊆ S such that f (C) = f (S) minimizing s∈C w s. (Taking f (C) = | ∪ s∈C s| gives weighted set cover.) 2 KEY RESULTS The greedy algorithm for weighted set cover builds a cover by repeatedly choosing a set s that minimize the weight w s divided by number of elements in s not yet covered by chosen sets. It stops and returns the chosen sets when they form a cover: greedy-set-cover(S, w) 1. Initialize C ← ∅. Define f (C). = | ∪ s∈C s|. 2. Repeat until f (C) = f (S): 3. Choose s ∈ S minimizing the price per element w s /[f (C ∪ {s}) − f (C)]. 4. Let H k denote k i=1 1/i ≈ ln k, where k is the largest set size. Theorem 1. The greedy algorithm returns a set cover of weight at most H k times the minimum weight of any cover.