Abstract
We study the complexity and approximability of Cut Packing and Cycle Packing. For Cycle Packing, we show that the problem is \( \mathcal{A}\mathcal{P}\mathcal{X} \)-hard but can be approximated within a factor of O(log n) by a simple greedy approach. Essentially the same approach achieves constant approximation for “dense” graphs. We show that both problems are \( \mathcal{N}\mathcal{P} \)-hard for planar graphs. For Cut Packing we show that, given a graph G the maximum cut packing is always between α(G) and 2α(G). We then derive new or improved polynomial-time algorithms for Cut Packing for special classes of graphs.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A.A. Ageev, A.V. Kostochka, Z. Szigeti, A Characterization of Seymour Graphs. J. Graph Theory 24 (1997) 357–364.
A.A. Ageev, On Finding the Maximum Number of Disjoint Cuts in Seymour Graphs. Proceedings of the 7th European Symposium on Algorithms (ESA’99), Lecture Notes in Comput. Sci., 1643, Springer, Berlin (1999) 490–497.
V. Bafna and P.A. Pevzner, Genome Rearrangements and Sorting by Reversals. SIAM J. on Computing 25 (1996) 272–289.
B.S. Baker, Approximation Algorithms for \( \mathcal{N}\mathcal{P} \)-Complete Problems on Planar Graphs. J. ACM 41 (1994) 153–180.
P. Berman, T. Fujito, On Approximation Properties of the Independent Set Problem for Low Degree Graphs. Theory of Computing Systems 32 (1999) 115–132.
P. Berman and M. Karpinski, On Some Tighter Inapproximability Results. ECCC Report No. 29, University of Trier (1998).
B. Bollobás, Extremal Graph Theory, Academic Press, New-York (1978).
R. Boppana, M.M. Halldórsson, Approximating Maximum Independent Sets by Excluding Subgraphs. Bit 32 (1992) 180–196.
A. Caprara, Sorting Permutations by Reversals and Eulerian Cycle Decompositions. SIAM J. on Discrete Mathematics 12 (1999) 91–110.
C.J. Colbourn, The Combinatorics of Network Reliability. Oxford University Press (1986).
G.A. Dirac, On Rigid Circuit Graphs. Abh. Math. Sem. Univ. Hamburg 25 (1961) 71–76.
P. Erdös and L. Pósa. On the Maximal Number of Disjoint Circuits of a Graph. Publ. Math. Debrecen 9 (1962) 3–12.
P. Erdös and H. Sachs. Regulare Graphen Gegebener Taillenweite mit Minimaler Knotenzahl. Wittenberg Math.-Natur. Reine 12 (1963) 251–257.
A. Frank, Conservative Weightings and Ear-Decompositions of Graphs. Combinatorica 13 (1993) 65–81.
H.N. Gabow and R.E. Tarjan, Faster Scaling Algorithms for General Matching Problems. J. A CM 38 (1991) 815–853.
M. Grotschel, L. Lovász, A. Schrijver, Geometric algorithms and combinatorial optimization, Second edition: Algorithms and Combinatorics, 2. Springer-Verlag, Berlin, (1993). ISBN: 3-540-56740-2.
J. Håstad, Clique is Hard to Approximate within n1-ε. Acta Mathematica 182 (2000) 105–142.
I. Holyer, The \( \mathcal{N}\mathcal{P} \)-Completeness of Some Edge-Partition Problems. SIAM J. on Computing 10 (1981) 713–717.
H.B. HuntIII, M.V. Marathe, V. Radhakrishnan, S.S. Ravi, D.J. Rosenkrantz and R.E. Stearns, A Unified Approach to Approximation Schemes for \( \mathcal{N}\mathcal{P} \)-and \( \mathcal{P}\mathcal{S}\mathcal{P}\mathcal{A}\mathcal{C}\mathcal{E} \)-Hard Problems for Geometric Graphs. Proceedings of the 2nd Euoropean Symposium on Algorithms (ESA’94), Lecture Notes in Comput. Sci., 855, Springer, Berlin (1994) 424–435.
J. Kececioglu and D. Sankoff, Exact and Approximation Algorithms for Sorting by Reversals, with Application to Genome Rearrangement. Algorithmica 13 (1995) 180–210.
D. Lichtenstein, Planar formulae and their uses. SIAM J. on Computing 11 (1982) 329–343.
L. Lovász, M.D. Plummer, Matching Theory, Akadémiai Kiadó (1986)
C.H. Papadimitriou and M. Yannakakis (1991), Optimization, Approximation, and Complexity Classes J. Comput. System Sci. 43 (1991) 425–440.
D.J. Rose, R.E. Tarjan and G.S. Lueker, Algorithmic Aspects of Vertex Elimination on Graphs. SIAM J. on Computing 5 (1976) 266–283.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Caprara, A., Panconesi, A., Rizzi, R. (2001). Packing Cycles and Cuts in Undirected Graphs. In: auf der Heide, F.M. (eds) Algorithms — ESA 2001. ESA 2001. Lecture Notes in Computer Science, vol 2161. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44676-1_43
Download citation
DOI: https://doi.org/10.1007/3-540-44676-1_43
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42493-2
Online ISBN: 978-3-540-44676-7
eBook Packages: Springer Book Archive