Abstract
We develop a new randomized method, random separation, for solving fixed-cardinality optimization problems on graphs, i.e., problems concerning solutions with exactly a fixed number k of elements (e.g., k vertices V′) that optimize solution values (e.g., the number of edges covered by V′). The key idea of the method is to partition the vertex set of a graph randomly into two disjoint sets to separate a solution from the rest of the graph into connected components, and then select appropriate components to form a solution. We can use universal sets to derandomize algorithms obtained from this method.
This new method is versatile and powerful as it can be used to solve a wide range of fixed-cardinality optimization problems for degree-bounded graphs, graphs of bounded degeneracy (a large family of graphs that contains degree-bounded graphs, planar graphs, graphs of bounded tree-width, and nontrivial minor-closed families of graphs), and even general 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.
References
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)
Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)
Cai, L.: Parameterized complexity of cardinality constrained optimization problems. A special issue of The Computer Journal on parameterized complexity (submitted, 2006)
Cai, L., Chan, S.M., Chan, S.O.: Research notes (2006)
Chen, J., Kanj, I., Jia, W.: Vertex cover: further observations and further improvements. J. Algorithms 41, 280–301 (2001)
Courcelle, B.: The monadic second-order logic of graphs I: recognisable sets of finite graphs. Inform. Comput. 85(1), 12–75 (1990)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Frick, M., Grohe, M.: Deciding first-order properties of locally tree-decomposable structures. J. of the ACM 48(6), 1184–1206 (2001)
Kleinberg, J., Tardos, E.: Algorithm Design. Pearson, London (2005)
Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: Proc. 36th Annual Symp. Foundations of Computer Science, pp. 182–191 (1995)
Robertson, N., Seymour, P.D.: Graph minors XIII: the disjoint paths problem. J. Comb. Ther. (B) 63(1), 65–110 (1995)
Schmidt, J.P., Siegel, A.: The spatial complexity of oblivious k-probe hash functions. SIAM J. Comp. 19, 775–786 (1990)
Seese, D.: Linear time computable problems and first-order descriptions. Mathematical Structures in Computer Science 6(6), 505–526 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cai, L., Chan, S.M., Chan, S.O. (2006). Random Separation: A New Method for Solving Fixed-Cardinality Optimization Problems. In: Bodlaender, H.L., Langston, M.A. (eds) Parameterized and Exact Computation. IWPEC 2006. Lecture Notes in Computer Science, vol 4169. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11847250_22
Download citation
DOI: https://doi.org/10.1007/11847250_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-39098-5
Online ISBN: 978-3-540-39101-2
eBook Packages: Computer ScienceComputer Science (R0)