Parameterized Algorithms for Boxicity
Abstract
In this paper we initiate an algorithmic study of Boxicity, a combinatorially well studied graph invariant, from the viewpoint of parameterized algorithms. The boxicity of an arbitrary graph G with the vertex set V(G) and the edge set E(G), denoted by box(G), is the minimum number of interval graphs on the same set of vertices such that the intersection of the edge sets of the interval graphs is E(G). In the Boxicity problem we are given a graph G together with a positive integer k, and asked whether the box(G) is at most k. The problem is notoriously hard and is known to be NP-complete even to determine whether the boxicity of a graph is at most two. This rules out any possibility of having an algorithm with running time |V(G)| O(f(k)), where f is an arbitrary function depending on k alone. Thus we look for other structural parameters like “vertex cover number” and “max leaf number” and see its effect on the problem complexity. In particular, we give an algorithm that given a vertex cover of size k finds box(G) in time \(2^{O(2^k k^2)}|V(G)|\). We also give a faster additive one approximation algorithm for finding box(G) that given a graph with vertex cover of size k runs in time \(2^{O(k^2 \log k)}|V(G)|\). Our next result is an additive two approximation algorithm for Boxicity when parameterized by the max leaf number running in time \(2^{O(k^3\log k)}|V(G)|^{O(1)}\). Our results are based on structural relationships between boxicity and the corresponding parameter and could be of independent interest.
Keywords
Span Tree Bipartite Graph Vertex Cover Parameterized Algorithm Interval GraphPreview
Unable to display preview. Download preview PDF.
References
- 1.Adiga, A., Bhowmick, D., Chandran, L.S.: Boxicity and poset dimension. In: Thai, T. (ed.) COCOON 2010. LNCS, vol. 6196, pp. 3–12. Springer, Heidelberg (2010)Google Scholar
- 2.Adiga, A., Bhowmick, D., Chandran, L.S.: The hardness of approximating the threshold dimension, boxicity and cubicity of a graph. DAM 158(16), 1719–1726 (2010)zbMATHGoogle Scholar
- 3.Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)Google Scholar
- 4.Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
- 5.Cai, L., Huang, X.: Fixed-parameter approximation: Conceptual framework and approximability results. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 96–108. Springer, Heidelberg (2006)CrossRefGoogle Scholar
- 6.Chandran, L.S., Das, A., Shah, C.D.: Cubicity, boxicity, and vertex cover. Disc. Math. 309, 2488–2496 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Chandran, L.S., Francis, M.C., Sivadasan, N.: Boxicity and maximum degree. J. Comb. Theory Ser. B 98(2), 443–445 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Chandran, L.S., Sivadasan, N.: Boxicity and treewidth. J. Comb. Theory Ser. B 97(5), 733–744 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Chen, Y., Grohe, M., Grüber, M.: On parameterized approximability. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 175–183. Springer, Heidelberg (2006)CrossRefGoogle Scholar
- 10.Cozzens, M.B.: Higher and multi-dimensional analogues of interval graphs, Ph.D. thesis. Department of Mathematics, Rutgers University, New Brunswick (1981)Google Scholar
- 11.Cozzens, M.B., Roberts, F.S.: Computing the boxicity of a graph by covering its complement by cointerval graphs. Disc. Appl. Math. 6, 217–228 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
- 13.Downey, R.G., Fellows, M.R., McCartin, C.: Parameterized approximation problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 121–129. Springer, Heidelberg (2006)CrossRefGoogle Scholar
- 14.Esperet, L.: Boxicity of graphs with bounded degree. European J. Combin. 30(5), 1277–1280 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Fellows, M.R., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F.A., Saurabh, S.: The complexity ecology of parameters: An illustration using bounded max leaf number. Theory Comput. Syst. 45(4), 822–848 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Fellows, M.R., Lokshtanov, D., Misra, N., Rosamond, F.A., Saurabh, S.: Graph layout problems parameterized by vertex cover. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 294–305. Springer, Heidelberg (2008)CrossRefGoogle Scholar
- 17.Fiala, J., Golovach, P.A., Kratochvíl, J.: Parameterized complexity of coloring problems: Treewidth versus vertex cover. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, Springer, Heidelberg (2009)Google Scholar
- 18.Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)zbMATHGoogle Scholar
- 19.Kleitman, D.J., West, D.B.: Spanning trees with many leaves. SJDM 4, 99–106 (1991)Google Scholar
- 20.Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Disc. Appl. Math. 52, 233–252 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Marx, D., Razgon, I.: Constant ratio fixed-parameter approximation of the edge multicut problem. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 647–658. Springer, Heidelberg (2009)CrossRefGoogle Scholar
- 22.Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
- 23.Roberts, F.S.: On the boxicity and cubicity of a graph. In: Recent Progresses in Combinatorics, pp. 301–310. Academic Press, New York (1969)Google Scholar
- 24.Scheinerman, E.R.: Intersection classes and multiple intersection parameters, Ph.D. thesis, Princeton University (1984)Google Scholar
- 25.Thomassen, C.: Interval representations of planar graphs. J. Comb. Theory Ser. B 40, 9–20 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
- 26.Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Alg. Disc. Math. 3(3), 351–358 (1982)MathSciNetzbMATHGoogle Scholar