Computing Cycle Covers without Short Cycles [chapter]

Markus Bläser, Bodo Siebert
2001 Lecture Notes in Computer Science  
A cycle cover of a graph is a spanning subgraph where each node is part of exactly one simple cycle. A k-cycle cover is a cycle cover where each cycle has length at least k. We call the decision problems whether a directed or undirected graph has a k-cycle cover k-DCC and k-UCC. Given a graph with edge weights one and two, Min-k-DCC and Min-k-UCC are the minimization problems of finding a k-cycle cover with minimum weight. We present factor 4/3 approximation algorithms for Min-k-DCC with
more » ... time O(n 5/2 ) (independent of k). Specifically, we obtain a factor 4/3 approximation algorithm for the asymmetric travelling salesperson problem with distances one and two and a factor 2/3 approximation algorithm for the directed path packing problem with the same running time. On the other hand, we show that k-DCC is N P-complete for k ≥ 3 and that Min-k-DCC has no PTAS for k ≥ 4, unless P = N P. Furthermore, we design a polynomial time factor 7/6 approximation algorithm for Min-k-UCC. As a lower bound, we prove that Min-k-UCC has no PTAS for k ≥ 12, unless P = N P. ⋆ supported by DFG research grant Re 672/3 9th Ann. European Symposium on Algorithms (ESA 2001) c Springer Input: a complete loopless directed graph G with edge weights one and two, an integer k ≥ 3. Output: a k-cycle cover of G.
doi:10.1007/3-540-44676-1_31 fatcat:flxpjeyzqza2dp7fwc42b3hqta