Cycle cover with short cycles

Authors:
Nicole Immorlica;Mohammad Mahdian;Vahab S. Mirrokni
Affiliations:
Computer Science and Artificial Intelligence Lab, MIT, Cambridge, MA;Computer Science and Artificial Intelligence Lab, MIT, Cambridge, MA;Computer Science and Artificial Intelligence Lab, MIT, Cambridge, MA
Venue:
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Year:
2005

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Abstract

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 [3,4] in weighted graphs with metric lengths on the edges and an O(ln k) approximation for the bounded cycle cover problem [9] 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. [3,4]) 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.