Parametric duality and kernelization: Lower bounds and upper bounds on kernel size

SIAM Journal on Computing

Volumn 37, Issue 4, 2007, Pages 1077-1106

  Chen, Jianer ^a   Fernau, Henning ^b   Kanj, Iyad A ^c   Xia, Ge ^d  

^a Texas A and M University   (United States)

^b UNIVERSITY OF TRIER   (Germany)

^c DEPAUL UNIVERSITY   (United States)

^d LAFAYETTE COLLEGE   (United States)

Author keywords

Dominating set; Independent set; Kernel; Parameterized algorithm; Planar graph; Vertex cover

Indexed keywords

COMBINATORIAL PROPERTIES; DOMINATING SET; DOMINATING SETS; EFFICIENT ALGORITHMS; HARD PROBLEMS; INDEPENDENT SET; INDEPENDENT SETS; KERNEL; KERNEL SIZE; KERNELIZATION; LOWER BOUNDS; NEW TECHNIQUES; PARAMETERIZED ALGORITHM; PARAMETERIZED COMPLEXITY; PARAMETERIZED PROBLEMS; PLANAR GRAPH; PLANAR VERTEX COVER; PRACTICAL ALGORITHMS; PROBLEM KERNEL; REDUCTION ALGORITHMS; UPPER AND LOWER BOUNDS; UPPER BOUNDS; VERTEX COVER; VERTEX COVER PROBLEMS;

ALGORITHMS; CHLORINE COMPOUNDS; CONTROL THEORY; DATA REDUCTION; TRANSIENTS;

SET THEORY;

EID: 49449114354     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/050646354     Document Type: Article

References (41)

1. F. ABU-KHZAM AND M. LANGSTON, A direct algorithm for the parameterized face cover problem, in Proceedings of the 1st International Workshop on Parameterized and Exact Computation, Lecture Notes in Comput. Sci. 3162, Springer, Berlin, 2004, pp. 213-222.
2. F. ABU-KHZAM, M. LANGSTON, AND P. SHANBHAG, A High-Performance Toolkit for Fast Exact Algorithms, invited talk to the Workshop on Fixed-Parameter Tractability, Ottawa, ON, Canada, 2003; available online at http://citeseer.ist.psu.edu/560785.html.
3. J. ALBER, N. BETZLER, AND R. NIEDERMEIER, Experiments on data reduction for optimal domination in networks, Ann. Oper. Res., 146 (2006), pp. 105-117.
4. J. ALBER, H. BODLAENDER, H. FERNAU, T. KLOKS, AND R. NIEDERMEIER, Fixed parameter algorithms for dominating set and related problems on planar graphs, Algorithmica, 33 (2002), pp. 461-493.
5. J. ALBER, M. FELLOWS, AND R. NIEDERMEIER, Polynomial-time data reduction for dominating set, J. ACM, 51 (2004), pp. 363-384.
6. J. ALBER, H. FERNAU, AND R. NIEDERMEIER, Parameterized complexity: Exponential speedup for planar graph problems, J. Algorithms, 52 (2004), pp. 26-56.
7. J. ALBER, H. FERNAU, AND R. NIEDERMEIER, Graph separators: A parameterized view, J. Comput. System Sci., 67 (2003), pp. 808-832.
8. B. BAKER, Approximation algorithms for NP-complete problems on planar graphs, J. ACM, 41 (1994), pp. 153-180.
9. R. BALASUBRAMANIAN, M. R. FELLOWS, AND V. RAMAN, An improved fixed parameter algorithm for vertex cover, Inform. Process. Lett., 65 (1998), pp. 163-168.
10. J. CHEETHAM, F. DEHNE, A. RAU-CHAPLIN, U. STEGE, AND P. J. TAILLON, Solving large FPT problems on coarse grained parallel machines, J. Comput. System Sci., 67 (2003), pp. 691-706.
11. J. CHEN AND I. KANJ, Constrained minimum vertex cover in bipartite graphs: Complexity and parameterized algorithms, J. Comput. System Sci., 67 (2003), pp. 833-847.
12. J. CHEN AND I. KANJ, Improved exact algorithms for MAX-SAT, Discrete Appl. Math., 142 (2004), pp. 17-27.
13. J. CHEN, I. KANJ, AND W. JIA, Vertex cover: Further observations and further improvement, J. Algorithms, 41 (2001), pp. 280-301.
14. J. CHEN, I. KANJ, AND G. XIA, Improved parameterized upper bounds for vertex cover, in Proceedings of the 31st International Symposium on the Mathematical Foundations of Computer Science, Lecture Notes in Comput. Sci. 4162, Springer, Berlin, 2006, pp. 238-249.
15. F. DEHNE, M. FELLOWS, H. FERNAU, E. PRIETO, AND F. ROSAMOND, Nonblocker: Parameterized algorithmes for MINIMUM DOMINATING SET, in Software Seminar SOFSEM, Lecture Notes in Comput. Sci. 3831, Springer, Berlin, 2006, pp. 237-245.
16. R. DIESTEL, Graph Theory, Springer, Berlin, 1997.
17. R. DOWNEY AND M. FELLOWS, Parameterized Complexity, Springer, New York, 1999.
18. R. DOWNEY, M. FELLOWS, AND U. STEGE, Parameterized complexity: A framework for systematically confronting computational intractability, in Contemporary Trends in Discrete Mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 49, R. Graham, J. Kratochvíl, J. Nešetřil, and F. Roberts, eds., AMS, Providence, RI, 1999, pp. 49-99.
19. M. FELLOWS, Parameterized complexity: The main ideas and connections to practical computing, in Electronic Notes in Theoretical Computer Science, Vol. 61, 2002.
20. M. FELLOWS, C. MCCARTIN, F. ROSAMOND, AND U. STEGE, Coordinatized kernels and catalytic reductions: An improved FPT algorithm for max leaf spanning tree and other problems, in FST TCS 2000, Lecture Notes in Comput. Sci. 1974, Springer, Berlin, 2000, pp. 240-251.
21. H. FERNAU, On parameterized enumeration, in Computing and Combinatorics, Lecture Notes in Comput. Sci. 2387, Springer, Berlin, 2002, pp. 564-573.
22. H. FERNAU, ROMAN Domination: A parameterized perspective, in Software Seminar SOFSEM 2006, Lecture Notes in Comput. Sci. 3831, Springer, Berlin, 2006, pp. 262-271.
23. H. FERNAU AND D. JUEDES, A geometric approach to parameterized algorithms for domination problems on planar graphs, in Mathematical Foundations of Computer Science 2004, Lecture Notes in Comput. Sci. 3153, Springer, Berlin, 2004, pp. 488-499.
24. H. FERNAU AND R. NIEDERMEIER, An efficient exact algorithm for constraint bipartite vertex cover, J. Algorithms, 38 (2001), pp. 374-410.
25. F. V. FOMIN AND D. M. THILIKOS, Dominating sets in planar graphs: Branch-width and exponential speed-up, in Proceedings of the Fourteenth ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, 2003, pp. 168-177.
26. F. V. FOMIN AND D. THILIKOS, Fast parameterized algorithms for graphs on surfaces: Linear kernel and exponential speed-up, in Automata, Languages and Programming (ICALP), Lecture Notes in Comput. Sci. 3142, Springer, Berlin, 2004, pp. 581-592.
27. M. GAREY AND D. JOHNSON, Computers and Intractability: A Guide to the Theory of NPCompleteness, Freeman, San Francisco, 1979.
28. J. GIMBEL AND C. THOMASSEN, Coloring triangle-free graphs with fixed size, Discrete Math., 219 (2000), pp. 275-277.
29. H. GRÖTZSCH, Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. MartinLuther-Univ. Halle-Wittenberg Ma.-Nat. Reihe, 8 (1958/1959), pp. 109-120.
30. F. HENGLEIN AND H. G. MAIRSON, The complexity of type inference for higher order typed lambda calculi, J. Funct. Programming, 4 (1994), pp. 435-477.
31. J. HOPCROFT AND R. TARJAN, Efficient planarity testing, J. ACM, 21 (1974), pp. 549-568.
32. R. HYAFIL AND R. L. RIVEST, Constructing optimal binary trees is NP-complete, Inform. Process. Lett., 5 (1976), pp. 15-17.
33. I. KANJ AND L. PERKOVIC̀, Improved parameterized algorithms for planar dominating set, in Mathematical Foundations of Computer Science 2002, Lecture Notes in Comput. Sci. 2420, Springer, Berlin, 2002, pp. 399-410.
34. S. KHOT AND V. RAMAN, Parameterized complexity of finding subgraphs with hereditary properties, Theoret. Comput. Sci., 289 (2002), pp. 997-1008.
35. O. LICHTENSTEIN AND A. PNEULI, Checking that finite-state concurrent programs satisfy their linear specification, in Proceedings of the 12th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages (POPL), 1985, pp. 97-107.
36. R. NIEDERMEIER AND P. ROSSMANITH, A general method to speed up fixed-parameter-tractable algorithms, Inform. Process. Lett., 73 (2000), pp. 125-129.
37. O. ORE, Theory of Graphs, Amer. Math. Soc. Colloq. Publ. XXXVIII, AMS, Providence, RI, 1962.
38. E. PRIETO AND C. SLOPER, Either/or: Using vertex cover structure in designing FPTalgorithms-the case of k-Internal Spanning Tree, in Algorithms and Data Structures (WADS), Lecture Notes in Comput. Sci. 2748, Springer, Berlin, 2003, pp. 474-483.
39. C. THOMASSEN, Grötzsch's 3-color theorem and its counterparts for the torus and the projective plane, J. Combin. Theory Ser. B, 62 (1994), pp. 268-279.
40. R. UEHARA, Probabilistic Algorithms and Complexity Classes, Ph.D. thesis, Department of Computer Science and Information Mathematics, The University of ElectroCommunications, Japan, 1998.
41. K. WEIHE, Covering trains by stations or the power of data reduction, in Proceedings of the Workshop on Algorithms and Experiments (ALEX '98), 1998, pp. 1-8; available online at http://rtm.science.unitn.it/ alex98/proceedings.html.