Approximation algorithms and hardness results for cycle packing problems

  • Authors:

  • Michael Krivelevich;Zeev Nutov;Mohammad R. Salavatipour;Jacques Verstraete Yuster;Raphael Yuster

  • Affiliations:

  • Tel Aviv University, Tel Aviv, Israel;Open University of Israel, Tel Aviv, Israel;University of Alberta, Edmonton, Canada;University of Waterloo, Canada;University of Haifa, Haifa, Israel

  • Venue:
  • ACM Transactions on Algorithms (TALG)
  • Year:
  • 2007

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Abstract

The cycle packing number νe(G) of a graph G is the maximum number of pairwise edge-disjoint cycles in G. Computing νe(G) is an NP-hard problem. We present approximation algorithms for computing νe(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(&sqrt;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 &sqrt;n-approximation algorithm. Finally, we give an O(n2/3)-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 νe(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 νe(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.