Greedy in Approximation Algorithms

The objective of this paper is to characterize classes of problems for which a greedy algorithm finds solutions provably close to optimum. To that end, we introduce the notion of k-extendible systems, a natural generalization of matroids, and show that a greedy algorithm is a 1 k -factor approximation for these systems. Many seemly unrelated problems fit in our framework, e.g.: b-matching, maximum profit scheduling and maximum asymmetric TSP. In the second half of the paper we focus on the maximum weight b-matching problem. The problem forms a 2-extendible system, so greedy gives us a 1 2 -factor solution which runs in O(m log n) time. We improve this by providing two linear time approximation algorithms for the problem: a 1 2 -factor algorithm that runs in O(bm) time, and a ` 2 3 − ǫ ́ -factor algorithm which runs in expected O ` bm log 1 ǫ ́