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 $O(\sqrt{\log n})$ [14,15]. 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 [1,9], we show that the undirected edge-disjoint cycle packing problem has an integrality gap of $\Omega(\frac{\sqrt{\log n}}{\log\log n})$ and furthermore it is quasi-NP-hard to approximate the edge-disjoint cycle problem within ratio of $O(\log^{\frac{1}{2}-\epsilon} n)$ for any constant ε> 0. The same results hold for the problem of packing vertex-disjoint cycles.