Approximation Algorithms and Hardness Results for Cycle Packing Problems

The cycle packing number nu(e)(G) of a graph G is the maximum number of pairwise edge-disjoint cycles in G. Computing nu(e)(G) is an NP-hard problem. We present approximation algorithms for computing nu(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 Theta(root log n), where n = vertical bar V(G)vertical bar. This improves upon the previous O(log n) upper bound for the approximation ratio of this algorithm. In the directed case we present a root 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 nu(e)(G) in directed graphs. Specifically, we prove a lower bound of Omega(log n/log log n) for the integrality gap of edge-disjoint cycle packing. We also show that it is quasi-NP-hard to approximate nu(e)(G) within a factor of O(log(1-epsilon) n) for any constant epsilon > 0. This improves upon the previously known APX-hardness result for this problem.