Subexponential Parameterized Algorithm for Minimum Fill-In
Abstract
The Minimum Fill-in problem is used to decide if a graph can be triangulated by adding at most $k$ edges. In 1994, Kaplan, Shamir, and Tarjan showed that the problem is solvable in time $\mathcal{O}(2^{\mathcal{O}({k})}+k^2 nm)$ on graphs with $n$ vertices and $m$ edges and thus is fixed parameter tractable. Here, we give the first subexponential parameterized algorithm solving Minimum Fill-in in time $\mathcal{O}(2^{\mathcal{O}(\sqrt{k}\log{k})} +k^2 nm)$. This substantially lowers the complexity of the problem. Techniques developed for Minimum Fill-in can be used to obtain subexponential parameterized algorithms for several related problems, including Minimum Chain Completion, Chordal Graph Sandwich, and Triangulating Colored Graph.
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References
1.
N. Alon, D. Lokshtanov, and S. Saurabh, Fast FAST, Automata, Languages, and Programming, Lecture Notes in Comput. Sci. 5555, Springer, New York, 2009, pp. 49--58.
2.
C. Beeri, R. Fagin, D. Maier, and M. Yannakakis, On the desirability of acyclic database schemes, J. ACM, 30 (1983), pp. 479--513.
3.
J. R. S. Blair and B. W. Peyton, An introduction to chordal graphs and clique trees, in Graph Theory and Sparse Matrix Computations, IMA Vol. Math. Appl. 56, Springer, New York, 1993, pp. 1--30.
4.
H. Bodlaender, P. Heggernes, and Y. Villanger, Faster parameterized algorithms for minimum fill-in, Algorithmica, 61 2011, pp. 817--838.
5.
H. L. Bodlaender, M. R. Fellows, M. T. Hallett, T. Wareham, and T. Warnow, The hardness of perfect phylogeny, feasible register assignment and other problems on thin colored graphs, Theoret. Comput. Sci., 244 (2000), pp. 167--188.
6.
H. L. Bodlaender, M. R. Fellows, and T. Warnow, Two strikes against perfect phylogeny, in Automata, Languages, and Programming, Lecture Notes in Comput. Sci. 623, Springer, New York, 1992, pp. 273--283.
7.
M. L. Bonet, C. A. Phillips, T. Warnow, and S. Yooseph, Constructing evolutionary trees in the presence of polymorphic characters, in Proceedings of STOC, ACM, 1996, pp. 220--229.
8.
V. Bouchitté and I. Todinca, Treewidth and minimum fill-in: Grouping the minimal separators, SIAM J. Comput., 31 (2001), pp. 212--232.
9.
V. Bouchitté and I. Todinca, Listing all potential maximal cliques of a graph, Theoret. Comput. Sci., 276 (2002), pp. 17--32.
10.
P. Buneman, A characterisation of rigid circuit graphs, Discrete Math., 9 (1974), pp. 205--212.
11.
L. Cai, Fixed-parameter tractability of graph modification problems for hereditary properties, Inform. Process. Lett., 58 (1996), pp. 171--176.
12.
L. S. Chandran and F. Grandoni, A linear time algorithm to list the minimal separators of chordal graphs, Discrete Math., 306 (2006), pp. 351--358.
13.
F. R. K. Chung and D. Mumford, Chordal completions of planar graphs, J. Combin. Theory Ser. B, 62 (1994), pp. 96--106.
14.
M. Cygan, private communication, 2011.
15.
T. A. Davis, Direct Methods for Sparse Linear Systems, Fundam. Algorithms 2, SIAM, Philadelphia, 2006.
16.
E. D. Demaine, F. V. Fomin, M. Hajiaghayi, and D. M. Thilikos, Subexponential parameterized algorithms on graphs of bounded genus and $H$-minor-free graphs, J. Assoc. Comput. Mach., 52 (2005), pp. 866--893.
17.
R. Diestel, Graph Theory, 3rd ed., Grad. Texts in Math. 173, Springer, Berlin, 2005.
18.
G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg, 25 (1961), pp. 71--76.
19.
R. G. Downey and M. R. Fellows, Parameterized complexity, Springer, New York, 1999.
20.
T. Feder, H. Mannila, and E. Terzi, Approximating the minimum chain completion problem, Inform. Process. Lett., 109 (2009), pp. 980--985.
21.
J. Flum and M. Grohe, Parameterized Complexity Theory, Texts Theoret. Comput. Sci. EATCS Ser., Springer, Berlin, 2006.
22.
F. V. Fomin and D. Kratsch, Exact Exponential Algorithms, Texts Theoret. Comput. Sci. EATCS Ser., Springer, Berlin, 2010.
23.
F. V. Fomin, D. Kratsch, I. Todinca, and Y. Villanger, Exact algorithms for treewidth and minimum fill-in, SIAM J. Comput., 38 (2008), pp. 1058--1079.
24.
F. V. Fomin and Y. Villanger, Treewidth computation and extremal combinatorics, in Automata, Languages, and Programming, Lecture Notes in Comput. Sci. 5125, Springer, New York, 2008, pp. 210--221.
25.
F. V. Fomin and Y. Villanger, Finding induced subgraphs via minimal triangulations, in STACS, LIPIcs 5, Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2010, pp. 383--394.
26.
D. R. Fulkerson and O. A. Gross, Incidence matrices and interval graphs, Pacific J. Math., 15 (1965), pp. 835--855.
27.
M. R. Garey and D. S. Johnson, Computers and Intractability, A Guide to the Theory of NP-Completeness, W.H. Freeman, New York, 1979.
28.
D. Geman, Random fields and inverse problems in imaging, in École d'été de Probabilités de Saint-Flour XVIII---1988, Lecture Notes in Math. 1427, Springer, Berlin, 1990, pp. 113--193.
29.
M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
30.
M. C. Golumbic, H. Kaplan, and R. Shamir, Graph sandwich problems, J. Algorithms, 19 (1995), pp. 449--473.
31.
P. Heggernes, Minimal triangulations of graphs: A survey, Discrete Math., 306 (2006), pp. 297--317.
32.
P. Heggernes, C. Paul, J. A. Telle, and Y. Villanger, Interval completion with few edges, in Proceedings of STOC, 2007, pp. 374--381.
33.
P. Heggernes, J. A. Telle, and Y. Villanger, Computing minimal triangulations in time ${O}( n^\alpha \log n) =o(n^{{2.376}})$, SIAM J. Discrete Math., 19 (2005), pp. 900--913.
34.
R. Impagliazzo, R. Paturi, and F. Zane, Which problems have strongly exponential complexity?, J. Comput. System Sci., 63 (2001), pp. 512--530.
35.
M. Jurdzinski, M. Paterson, and U. Zwick, A deterministic subexponential algorithm for solving parity games, SIAM J. Comput., 38 (2008), pp. 1519--1532.
36.
S. Kannan and T. Warnow, Triangulating $3$-colored graphs, SIAM J. Discrete Math., 5 (1992), pp. 249--258.
37.
H. Kaplan, R. Shamir, and R. E. Tarjan, Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping, in Proceedings of FOCS, IEEE, 1994, pp. 780--791.
38.
H. Kaplan, R. Shamir, and R. E. Tarjan, Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs, SIAM J. Comput., 28 (1999), pp. 1906--1922.
39.
T. Kloks, D. Kratsch, and J. Spinrad, On treewidth and minimum fill-in of asteroidal triple-free graphs, Theoret. Comput. Sci., 175 (1997), pp. 309--335.
40.
D. Kratsch and J. Spinrad, Between ${O}(nm)$ and ${O}(n^{\alpha})$, SIAM J. Comput., 36 (2006), pp. 310--325.
41.
S. L. Lauritzen and D. J. Spiegelhalter, Local computations with probabilities on graphical structures and their application to expert systems, J. Royal Statist. Soc. B, 50 (1988), pp. 157--224.
42.
D. Lokshtanov, D. Marx, and S. Saurabh, Lower bounds based on the Exponential Time Hypothesis, Bull. EATCS 105, 2011, pp. 41--72.
43.
H. Mannila and E. Terzi, Nestedness and segmented nestedness, in Proceedings of KDD '07, ACM, New York, 2007, pp. 480--489.
44.
A. Natanzon, R. Shamir, and R. Sharan, A polynomial approximation algorithm for the minimum fill-in problem, in Proceedings of STOC, 1998, pp. 41--47.
45.
A. Natanzon, R. Shamir, and R. Sharan, A polynomial approximation algorithm for the minimum fill-in problem, SIAM J. Comput., 30 (2000), pp. 1067--1079.
46.
T. Ohtsuki, L. K. Cheung, and T. Fujisawa, Minimal triangulation of a graph and optimal pivoting ordering in a sparse matrix, J. Math. Anal. Appl., 54 (1976), pp. 622--633.
47.
S. Parter, The use of linear graphs in Gauss elimination, SIAM Review, 3 (1961), pp. 119--130.
48.
B. Patterson and W. Atmar, Nested subsets and the structure of insular mammalian faunas and archipelagos, Biological J. Linnean Society, 28 (1986).
49.
D. J. Rose, A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations, in Graph Theory and Computing, R. C. Read, ed., Academic Press, New York, 1972, pp. 183--217.
50.
D. J. Rose, R. E. Tarjan, and G. S. Lueker, Algorithmic aspects of vertex elimination on graphs, SIAM J. Comput., 5 (1976), pp. 266--283.
51.
Y. Villanger, P. Heggernes, C. Paul, and J. A. Telle, Interval completion is fixed parameter tractable, SIAM J. Comput., 38 (2009), pp. 2007--2020.
52.
V. Vassilevska Williams, Multiplying matrices faster than Coppersmith-Winograd, in Proceedings of STOC, ACM, 2012, pp. 887--898.
53.
S. Wong, D. Wu, and C. Butz, Triangulation of bayesian networks: A relational database perspective, in Rough Sets and Current Trends in Computing, Lecture Notes in Comput. Sci. 2475, Springer, New York, 2002, pp. 950--950.
54.
M. Yannakakis, Computing the minimum fill-in is NP-complete, SIAM J. Algebraic Discrete Methods, 2 (1981), pp. 77--79.
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© 2013, Society for Industrial and Applied Mathematics.
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Submitted: 3 November 2011
Accepted: 11 September 2012
Published online: 5 December 2013
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