Approximation algorithms for cycle packing problems

The cycle packing number v_c(G) of a graph G is the maximum number of pairwise edge-disjoint cycles in G. Computing v_c(G) is an NP-hard problem. We present approximation algorithms for computing v _c(G) in both the undirected and directed cases. In the undirected case we analyze the modified greedy algorithm suggested in [4] and show that it has approximation ratio O(√log n) where n = |V(G)|, and this is tight. This improves upon the previous O(log n) upper bound for the approximation ratio of this algorithm. In the directed case we present a √n-approximation algorithm. Finally, we give an O(n^2/3)-approximation algorithm for the problem of rinding a maximum number of edge-disjoint cycles that intersect a specified subset S of vertices. Our approximation ratios are the currently best known ones and, in addition, provide bounds on the integrality gap of standard LP-relaxations to these problems.