# On Approximating Restricted Cycle Covers

• Bodo Manthey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3879)

## Abstract

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.

## Keywords

Approximation Algorithm Directed Graph Undirected Graph Edge Weight Travel Salesman Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)
2. 2.
Alimonti, P., Kann, V.: Some APX-completeness results for cubic graphs. Theoret. Comput. Sci. 237(1–2), 123–134 (2000)
3. 3.
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (1999)
4. 4.
Bläser, M.: Approximationsalgorithmen für Graphüberdeckungsprobleme. Habilitationsschrift, Institut für Theoretische Informatik, Universität zu Lübeck, Lübeck, Germany (2002)Google Scholar
5. 5.
Bläser, M., Manthey, B.: Approximating maximum weight cycle covers in directed graphs with weights zero and one. Algorithmica 42(2), 121–139 (2005)
6. 6.
Bläser, M., Manthey, B., Sgall, J.: An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality. J. Discrete Algorithms (to appear)Google Scholar
7. 7.
Bläser, M., Shankar Ram, L., Sviridenko, M.I.: Improved approximation algorithms for metric maximum ATSP and maximum 3-cycle cover problems. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 350–359. Springer, Heidelberg (2005)
8. 8.
Bläser, M., Siebert, B.: Computing cycle covers without short cycles. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 368–379. Springer, Heidelberg (2001)
9. 9.
Cornuéjols, G.P., Pulleyblank, W.R.: A matching problem with side conditions. Discrete Math. 29(2), 135–159 (1980)
10. 10.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)
11. 11.
Hartvigsen, D.: An Extension of Matching Theory. PhD thesis, Department of Mathematics, Carnegie Mellon University (1984)Google Scholar
12. 12.
Hassin, R., Rubinstein, S.: On the complexity of the k-customer vehicle routing problem. Oper. Res. Lett. 33(1), 71–76 (2005)
13. 13.
Hell, P., Kirkpatrick, D.G., Kratochvíl, J., Kríz, I.: On restricted two-factors. SIAM J. Discrete Math. 1(4), 472–484 (1988)
14. 14.
Vornberger, O.: Easy and hard cycle covers. Technical report, Universität/ Gesamthochschule Paderborn (1980)Google Scholar