Disjoint Cycles: Integrality Gap, Hardness, and Approximation

In the edge-disjoint cycle packing problem we are given a graph G and we have to find a largest set of edge-disjoint cycles in G. The problem of packing vertex-disjoint cycles in G is defined similarly. The best approximation algorithms for edge-disjoint cycle packing are due to Krivelevich et al. [16], where they give an $O\sqrt{\rm log n}$-approximation for undirected graphs and an $O(\sqrt{n})$-approximation for directed graphs. They also conjecture that the problem in directed case has an integrality gap of $\Omega(\sqrt{\rm n})$. No non-trivial lower bound is known for the integrality gap of this problem. Here we show that both problems of packing edge-disjoint and packing vertex-disjoint cycles in a directed graph have an integrality gap of $\Omega(\frac{log n}{log log n})$. This is the first super constant lower bound for the integrality gap of these problems. We also prove that both problems are quasi-NP-hard to approximate within a factor of Ω(log1−− en), for any e > 0. For the problem of packing vertex-disjoint cycles, we give the first approximation algorithms with ratios O(log n) (for undirected graphs) and $O(\sqrt{n})$ (for directed graphs). Our algorithms work for the more general case where we have a capacity cv on every vertex v and we are seeking a largest set $\mathcal{C}$ of cycles such that at most cv cycles of $\mathcal{C}$ contain v.