Approximation algorithms and hardness results for cycle packing problems

The cycle packing number ν_e(G) of a graph G is the maximum number of pairwise edge-disjoint cycles in G. Computing ν_e(G) is an NP-hard problem. We present approximation algorithms for computing ν_e(G) in both undirected and directed graphs. In the undirected case we analyze a variant of the modified greedy algorithm suggested by Caprara et al. [2003] and show that it has approximation ratio Θ(√log n), where n = |V(G)|. 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 finding a maximum number of edge-disjoint cycles that intersect a specified subset S of vertices. We also study generalizations of these problems. Our approximation ratios are the currently best-known ones and, in addition, provide upper bounds on the integrality gap of standard LP-relaxations of these problems. In addition, we give lower bounds for the integrality gap and approximability of ν_e(G) in directed graphs. Specifically, we prove a lower bound of Ω(log n/loglog n) for the integrality gap of edge-disjoint cycle packing. We also show that it is quasi-NP-hard to approximate ν_e(G) within a factor of O(log¹ − ε n) for any constant ε > 0. This improves upon the previously known APX-hardness result for this problem.