Minimum-weight cycle covers and their approximability
Discrete Applied Mathematics
Two Approximation Algorithms for ATSP with Strengthened Triangle Inequality
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
An improved approximation algorithm for TSP with distances one and two
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
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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 $L$. 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 $\mathsf{APX}$-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 within a factor of $2$ for undirected graphs and within a factor of $8/3$ in the case of directed graphs. This holds for arbitrary sets $L$.