Approximability of Packing Disjoint Cycles

Abstract

Given a graph G, the edge-disjoint cycle packing problem is to find the largest set of cycles of which no two share an edge. For undirected graphs, the best known approximation algorithm has ratio $O(\sqrt{\log n})$ (Krivelevich et al. in ACM Trans. Algorithms, 2009, to appear). In fact, they proved the same upper bound for the integrality gap of this problem by presenting a simple greedy algorithm. Here we show that this is almost best possible. By modifying integrality gap and hardness results for the edge-disjoint paths problem (Andrews et al. in Proc. of 46th IEEE FOCS, pp. 226–244, 2005; Chuzhoy and Khanna in New hardness results for undirected edge disjoint paths. Manuscript, 2005), we show that the undirected edge-disjoint cycle packing problem is quasi-NP-hard to approximate within ratio of $O(\log^{\frac{1}{2}-\epsilon}n)$ for any constant e>0. The same result holds for the problem of packing vertex-disjoint cycles.