Abstract
The width measure treedepth, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. We present an algorithm which—given as input an n-vertex graph, a tree decomposition of width w, and an integer t—decides whether the input graph has treedepth at most t in time 2O(wt) ·n. We use this to construct further algorithms which do not require a tree decomposition as part of their input: A simple algorithm which decides treedepth in linear time for a fixed t, thus answering an open question posed by Ossona de Mendez and Nešetřil as to whether such an algorithm exists, a fast algorithm with running time \(2^{O(t^2)} \cdot n\) and an algorithm for chordal graphs with running time 2O(t logt)·n.
Research funded by DFG-Project RO 927/13-1 “Pragmatic Parameterized Algorithms”.
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Bodlaender, H.L., Deogun, J.S., Jansen, K., Kloks, T., Kratsch, D., Müller, H., Tuza, Z.: Rankings of graphs. SIAM Journal of Discrete Mathematics 11(1), 168–181 (1998)
Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: A O(ck n) 5-approximation algorithm for treewidth. CoRR, abs/1304.6321 (2013)
Bodlaender, H.L., Gilbert, J.R., Hafsteinsson, H., Kloks, T.: Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms 18(2), 238–255 (1995)
Bodlaender, H.L., Kratsch, D.: Personal communication (2014)
Courcelle, B.: The Monadic Second-Order Theory of Graphs. I. Recognizable Sets of Finite graphs. Information and Computation 85, 12–75 (1990)
Deogun, J.S., Kloks, T., Kratsch, D., Müller, H.: On vertex ranking for permutations and other graphs. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds.) STACS 1994. LNCS, vol. 775, pp. 747–758. Springer, Heidelberg (1994)
Dereniowski, D., Nadolski, A.: Vertex rankings of chordal graphs and weighted trees. Information Processing Letters 98, 96–100 (2006)
Diestel, R.: Graph Theory, 4th edn. Springer, Heidelberg (2010)
Duff, I.S., Reid, J.K.: The multifrontal solution of indefinite sparse symmetric linear equations. ACM Transactions on Mathematical Software 9, 302–325 (1983)
Fomin, F.V., Giannopoulou, A.C., Pilipczuk, M.: Computing tree-depth faster than 2n. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 137–149. Springer, Heidelberg (2013)
Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. Annals of Pure and Applied Logic 130(1-3), 3–31 (2004)
Katchalski, M., McCuaig, W., Seager, S.: Ordered colourings. Discrete Mathematics 142(1-3), 141–154 (1995)
Kaya, K., Uçar, B.: Constructing elimination trees for sparse unsymmetric matrices. SIAM Journal on Matrix Analysis and Applications 34(2), 345–354 (2013)
Leiserson, C.E.: Area-efficient graph layouts (for VLSI). In: FOCS, pp. 270–281 (1980)
Liu, J.W.H.: The role of elimination trees in sparse factorization. SIAM Journal on Matrix Analysis and Applications 11(1), 134–172 (1990)
Lokshtanov, D., Marx, D., Saurabh, S.: Known algorithms on graphs on bounded treewidth are probably optimal. In: Randall, D. (ed.) Proc. of 22nd SODA, pp. 777–789. SIAM (2011)
Nešetřil, J., Ossona de Mendez, P.: Grad and classes with bounded expansion I. Decompositions. European Journal of Combinatorics 29(3), 760–776 (2008)
Nešetřil, J., Ossona de Mendez, P.: Sparsity: Graphs, Structures, and Algorithms. Algorithms and Combinatorics, vol. 28. Springer, Heidelberg (2012)
Pothen, A.: The complexity of optimal elimination trees. Technical Report CS-88-13, Pennsylvannia State University (1988)
Pothen, A., Simon, H.D., Liou, K.-P.: Partitioning sparse matrices with eigenvectors of graphs. SIAM Journal of Matrix Analysis and Applications 11(3), 430–452 (1990)
Robertson, N., Seymour, P.D.: Graph minors XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B 63, 65–110 (1995)
Schäffer, A.A.: Optimal node ranking of trees in linear time. Information Processing Letters 33(2), 91–96 (1989)
Spielman, D.A., Teng, S.-H.: Spectral partitioning works: Planar graphs and finite element meshes. In: FOCS, pp. 96–105 (1996)
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Reidl, F., Rossmanith, P., Villaamil, F.S., Sikdar, S. (2014). A Faster Parameterized Algorithm for Treedepth. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_77
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