SIAM Journal on Discrete Mathematics
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Some APX-completeness results for cubic graphs
Theoretical Computer Science
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computing Cycle Covers without Short Cycles
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality
Journal of Discrete Algorithms
Improved approximation algorithms for metric maximum ATSP and maximum 3-cycle cover problems
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
On the complexity of the k-customer vehicle routing problem
Operations Research Letters
Minimum-weight cycle covers and their approximability
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
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
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A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. A special case of L-cycle covers are k-cycle covers for k∈ℕ, where the length of each cycle must be at least k. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges. We come close to settling the complexity and approximability of computing L-cycle covers. On the one hand, we show that for almost all L, computing L-cycle covers of maximum weight in directed and undirected graphs is APX-hard and NP-hard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing L-cycle covers of maximum weight can be approximated with factor 2.5 for undirected graphs and with factor 3 in the case of directed graphs. Finally, we show that 4-cycle covers of maximum weight in graphs with edge weights zero and one can be computed in polynomial time. As a by-product, we show that the problem of computing minimum vertex covers in λ-regular graphs is APX-complete for every λ≥3.