Cycle Cover with Short Cycles

Cycle covering is a well-studied problem in computer science. In this paper, we develop approximation algorithms for variants of cycle covering problems which bound the size and/or length of the covering cycles. In particular, we give a (1 + ln 2)-approximation for the lane covering problem [4, 5] in weighted graphs with metric lengths on the edges and an O(ln k) approximation for the bounded cycle cover problem [11] with cycle-size bound k in uniform graphs. Our techniques are based on interpreting a greedy algorithm (proposed and empirically evaluated by Ergun et al. [4, 5]) as a dual-fitting algorithm. We then find the approximation factor by bounding the solution of a factor-revealing non-linear program. These are the first non-trivial approximation algorithms for these problems. We show that our analysis is tight for the greedy algorithm, and change the process of the dual-fitting algorithm to improve the factor for small cycle bounds. Finally, we prove that variants of the cycle cover problem which bound cycle size or length are APX-hard.