Two Approximation Algorithms for 3-Cycle Covers

A cycle cover of a directed graph is a collection of node disjoint cycles such that every node is part of exactly one cycle. A k-cycle cover is a cycle cover in which every cycle has length at least k. While deciding whether a directed graph has a 2-cycle cover is solvable in polynomial time, deciding whether it has a 3-cycle cover is already NP-complete. Given a directed graph with nonnegative edge weights, a maximum weight 2-cycle cover can be computed in polynomial time, too. We call the corresponding optimization problem of finding a maximum weight 3-cycle cover Max-3-DCC. In this paper we present two polynomial time approximation algorithms for Max-3-DCC. The heavier of the 3-cycle covers computed by these algorithms has at least a fraction of 3 5 − ǫ, for any ǫ > 0, of the weight of a maximum weight 3-cycle cover. As a lower bound, we prove that Max-3-DCC is APX-complete, even if the weights fulfil the triangle inequality.