Approximation algorithms and hardness results for cycle packing problems
Michael Krivelevich, Zeev Nutov, Mohammad R. Salavatipour, Jacques Verstraëte, Raphael Yuster · ACM Transactions on Algorithms · 2007
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 (log 1 − ε n ) for any constant ε > 0. This improves upon the previously known APX-hardness result for this problem.