Exact algorithms for treewidth and minimum fill-in

SIAM Journal on Computing

Volumn 38, Issue 3, 2008, Pages 1058-1079

  Fomin, Fedor V ^a   Kratsch, Dieter ^b   Todinca, Ioan ^c   Villanger, Yngve ^a  

^a UNIVERSITY OF BERGEN   (Norway)

^b UNIVERSITÉ DE LORRAINE   (France)

^c UNIVERSITÉ D'ORLÉANS   (France)

Author keywords

Exact exponential algorithm; Fill in; Minimal separators; Minimal triangulation; Potential maximal clique; Treewidth

Indexed keywords

SEPARATORS; TREES (MATHEMATICS);

EXACT EXPONENTIAL ALGORITHM; FILL-IN; MINIMAL SEPARATORS; MINIMAL TRIANGULATION; POTENTIAL MAXIMAL CLIQUE; TREEWIDTH;

GRAPH THEORY;

EID: 55249107831     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/050643350     Document Type: Article

References (48)

1. E. AMIR, Efficient approximation for triangulation of minimum treewidth, in Proceedings of the Seventeenth Conference in Uncertainty in Artificial Intelligence (UAI-2001), Morgan Kaufmann, San Francisco, CA, 2001, pp. 7-15.
2. S. ARNBORG, D. G. CORNEIL, AND A. PROSKUROWSKI, Complexity of finding embeddings in a k-tree, SIAM J. Alg. Disc. Meth., 8 (1987), pp. 277-284.
3. n), J. Algorithms, 54 (2005), pp. 168-204.
4. A. B ERRY, J. P. BORDAT, AND O. COGIS, Generating all the minimal separators of a graph, Internat. J. Found. Comput. Sci., 11 (2000), pp. 397-403.
5. A. BJÖRKLUND AND T. HUSFELDT, Inclusion-exclusion algorithms for counting set partitions, in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, Washington, DC, 2006, pp. 575-582.
6. H. L. BODLAENDER, A linear-time algorithm for finding tree-decompositions of small treewidth, SIAM J. Comput., 25 (1996), pp. 1305-1317.
7. H. L. BODLAENDER, A partial k-arboretum of graphs with bounded treewidth, Theoret. Comput. Sci., 209 (1998), pp. 1-45.
8. H. L. BODLAENDER AND F. V. FOMIN, Tree decompositions with small cost, Discrete Appl. Math., 145 (2005), pp. 143-154.
9. H. L. BODLAENDER, F. V. FOMIN, A. M. C. A. KOSTER, D. KRATSCH, AND D. M. THILIKOS, On exact algorithms for treewidth, in Algorithms-ESA 2006, Lecture Notes in Comput. Sci. 4168, Springer. Berlin, 2006, pp. 672-683.
10. H. L. BODLAENDER, J. R. GILBERT, H. HAFSTEINSSON, AND T. KLOKS, Approximating treewidth, paihwidth, frontsize, and shortest elimination tree, J. Algorithms, 18 (1995), pp. 238-255.
11. V. BOUCHITTÉ, D. KRATSCH, H. MÜLLER, AND I. TODINCA, On treewidth approximations, Discrete Appl. Math., 136 (2004), pp. 183-196.
12. V. BOUCHITTÉ, F. MAZOIT, AND I. TODINCA, Chordal embeddings of planar graphs, Discrete Math., 273 (2003), pp. 85-102.
13. V. BOUCHITTÉ AND I. TODINCA, Treewidth and minimum fill-in: Grouping the minimal separators, SIAM J. Comput., 31 (2001), pp. 212-232.
14. V. BOUCHITTÉ AND I. TODINCA, Listing all potential maximal cliques of a graph, Theoret. Comput. Sci., 276 (2002), pp. 17-32.
15. T. BRUEGGEMANN AND W. KERN, An improved deterministic local search algorithm for 3-SAT, Theoret. Comput. Sci., 329 (2004), pp. 303-313.
16. L. CAI, Fixed-parameter tractability of graph modification problems for hereditary properties, Inform. Process. Lett., 58 (1996), pp. 171-176.
17. D. G. CORNEIL, S. OLARIU, AND L. STEWART, Asteroidal triple-free graphs, SIAM J. Discrete Math., 10 (1997), pp. 399-430.
18. D. G. CORNEIL, S. OLARIU, AND L. STEWART, Linear time algorithms for dominating pairs in asteroidal triple-free graphs, SIAM J. Comput., 28 (1999), pp. 1284-1297.
19. R. G. DOWNEY AND M. R. FELLOWS, Parameterized Complexity, Springer, New York, 1999.
20. U. FEIGE, Coping with the NP-hardness of the graph bandwidth problem, in Algorithm Theory-SWAT 2000, Lecture Notes in Comput. Sci. 1851, Springer, Berlin, 2000, pp. 10-19.
21. U. FEIGE, M. HAJIAGHAYI, AND J. R. LEE, Improved approximation algorithms for minimum-weight vertex separators, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC '05), ACM Press, New York, 2005, pp. 563-572.
22. F. V. FOMIN, P. FRAIGNIAUD, AND N. NISSE, Nondeterministic graph searching: From path-width to treewidth, in Mathematical Foundations of Computer Science 2005, Lecture Notes in Comput. Sci. 3618, Springer, Berlin, 2005, pp. 364-375.
23. F. V. FOMIN, F. GRANDONI, AND D. KRATSCH, Measure and conquer: Domination-a case study, in Automata, Languages and Programming (ICALP 2005), Lecture Notes in Comput. Sci. 3580, Springer, Berlin, 2005, pp. 191-203.
24. F. V. FOMIN, F. GRANDONI, AND D. KRATSCH, Some new techniques in design and analysis of exact (exponential) algorithms, Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 87 (2005), pp. 47-77.
25. F. V. FOMIN, D. KRATSCH AND I. TODINCA, Exact (exponential) algorithms for treewidth and minimum fill-in, in Automata, Languages and Programming (ICALP 2004), Lecture Notes in Comput. Sci. 3142, Springer, Berlin, 2004, pp. 568-580.
26. F. V. FOMIN AND D. M. THILIKOS, New upper bounds on the decomposability of planar graphs, J. Graph Theory, 51 (2006), pp. 53-81.
27. M. C. GOLUMBIC, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
28. M. HELD AND R. M. KARP, A dynamic programming approach to sequencing problems, J. Soc. Indust. Appl. Math., 10 (1962), pp. 196-210.
29. E. HOROWITZ AND S. SAHNI, Commuting partitions with applications to the knapsack problem, J. Assoc. Comput. Mach., 21 (1974), pp. 277-292.
30. R. IMPAGLIAZZO, R. PATURI, AND F. ZANE, Which problems have strongly exponential complexity, J. Comput. System Sci., 63 (2001), pp. 512-530.
31. K. IWAMA, Worst-case upper bounds for k-SAT. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 82 (2004), pp. 61-71.
32. 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.
33. 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.
34. n) algorithm for graph coloring and other partitioning problems via inclusion-exclusion, in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, Washington, DC, 2006, pp. 583-590.
35. E. L. LAWLER, A note on the complexity of the chromatic number problem, Inform. Process. Lett., 5 (1976), pp. 66-67.
36. n steps, Discrete Appl. Math., 10 (1985), pp. 287-295.
37. A. PARRA AND P. SCHEFFLER, Characterizations and algorithmic applications of chordal graph embeddings, Discrete Appl. Math., 79 (1997), pp. 171-188.
38. N. ROBERTSON AND P. D. SEYMOUR, Graph minors. II. Algorithmic aspects of tree-width, J. Algorithms, 7 (1986), pp. 309-322.
39. N. ROBERTSON AND P. D. SEYMOUR, Graph minors. X. Obstructions to tree-decomposition, J. Combin. Theory Ser. B, 52 (1991), pp. 153-190.
40. J. M. ROBSON, Algorithms for maximum independent sets, J. Algorithms, 7 (1986), pp. 425-44.
41. U. SCHÖNING, Algorithmics in exponential time, in STACS 2005, Lecture Notes in Comput. Sci. 3404, Springer, Berlin, 2005, pp. 36-43.
42. P. D. SEYMOUR AND R. THOMAS, Call routing and the ratcatcher, Combinatorica, 14 (1994), pp. 217-241.
43. R. E. TARJAN AND A. E. TROJANOWSKI, Finding a maximum independent set, SIAM J. Comput., 6 (1977), pp. 537-546.
44. Y. VILLANGER, Improved exponential-time algorithms for treewidth and minimum fill-in, in LATIN 2006: Theoretical Informatics, Lecture Notes in Comput. Sci. 3887, Springer, Berlin, 2006, pp. 800-811.
45. R. WILLIAMS, A new algorithm for optimal constraint satisfaction and its implications, in Automata, Languages and Programming (ICALP 2004), Lecture Notes in Comput. Sci. 3142, Springer, Berlin, 2004, pp. 1227-1237.
46. G. WOEGINGER, Exact algorithms for NP-hard problems: A survey, in Combinatorial Optimization-Eureka, You Shrink!, Lecture Notes in Comput. Sci. 2570, Springer, Berlin, 2003, pp. 185-207.
47. G. WOEGINGER, Space and time complexity of exact algorithms: Some open problems, in IWPEC 2004, Lecture Notes in Comput. Sci. 3162, Springer, Berlin, 2004, pp. 281-290.
48. M. YANNAKAKIS, Computing the minimum fill-in is NP-complete, SIAM J. Alg. Disc. Meth., 2 (1981), pp. 77-79.