Combinatorial optimization: algorithms and complexity
Combinatorial optimization: algorithms and complexity
An approximation algorithm for the asymmetric travelling salesman problem with distances one and two
Information Processing Letters
The traveling salesman problem with distances one and two
Mathematics of Operations Research
Handbook of combinatorics (vol. 1)
Handbook of combinatorics (vol. 1)
Some optimal inapproximability results
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Introduction To Automata Theory, Languages, And Computation
Introduction To Automata Theory, Languages, And Computation
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Square-Free 2-Factor Problem in Bipartite Graphs
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
Long tours and short superstrings
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Two Approximation Algorithms for 3-Cycle Covers
APPROX '02 Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization
Approximation algorithms for restricted cycle covers based on cycle decompositions
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
On approximating restricted cycle covers
WAOA'05 Proceedings of the Third international conference on Approximation and Online Algorithms
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
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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 running time O(n5/2) (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 NP-complete for k ≥ 3 and that Min-k-DCC has no PTAS for k ≥ 4, unless P = NP. 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 = NP.