Abstract
Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable obstruction sets and efficient order tests is not just one way of obtaining strongly uniform FPT algorithms, but that all of FPT may be captured in this way. Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial size. Consequently we investigate the interplay between the sizes of problem kernels and the sizes of the elements of such obstruction sets, obtaining several examples of how results in one area yield new insights in the other. We show how exponential-size minor-minimal obstructions for pathwidth k form the crucial ingredient in a novel or-cross-composition for k-Pathwidth, complementing the trivial and-composition that is known for this problem. In the other direction, we show that or-cross-compositions into a parameterized problem can be used to rule out the existence of efficiently generated quasi-orders on its instances that characterize the no-instances by polynomial-size obstructions.
- Faisal N. Abu-Khzam, Michael R. Fellows, Michael A. Langston, and W. Henry Suters. 2007. Crown structures for vertex cover kernelization. Theory Comput. Syst. 41, 3, 411--430. Google Scholar
Digital Library
- Stefan Arnborg, Derek G. Corneil, and Andrzej Proskurowski. 1987. Complexity of finding embeddings in a k-tree. SIAM J. Algebra. Discr. 8, 277--284. Google Scholar
Digital Library
- Hans L. Bodlaender. 1998. A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209, 1--2, 1--45. Google Scholar
Digital Library
- Hans L. Bodlaender. 2009. Kernelization: New upper and lower bound techniques. In Proceedings of 4th IWPEC. 17--37. Google Scholar
Digital Library
- Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. 2009a. On problems without polynomial kernels. J. Comput. Syst. Sci. 75, 8, 423--434. Google Scholar
Digital Library
- Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. 2009b. (Meta) Kernelization. In Proceedings of 50th FOCS. 629--638. Google Scholar
Digital Library
- Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. 2011. Cross-composition: a new technique for kernelization lower bounds. In Proceedings of 28th STACS. 165--176.Google Scholar
- Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. 2012. Kernel bounds for structural parameterizations of pathwidth. In Proceedings of 13th SWAT. 352--363. Google Scholar
Digital Library
- Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. 2013. Preprocessing for treewidth: A combinatorial analysis through kernelization. SIAM J. Discrete Math. 27, 4, 2108--2142.Google Scholar
Cross Ref
- Hans L. Bodlaender and Ton Kloks. 1996. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algor. 21, 2, 358--402. Google Scholar
Digital Library
- Hans L. Bodlaender and Rolf H. Möhring. 1993. The pathwidth and treewidth of cographs. SIAM J. Discrete Math. 6, 181--188. Google Scholar
Digital Library
- Kevin Cattell, Michael J. Dinneen, Rodney G. Downey, Michael R. Fellows, and Michael A. Langston. 2000. On computing graph minor obstruction sets. Theor. Comput. Sci. 233, 107--127. Google Scholar
Digital Library
- Jianer Chen, Iyad A. Kanj, and Weijia Jia. 2001. Vertex cover: Further observations and further improvements. J. Algor. 41, 2, 280--301. Google Scholar
Digital Library
- Jianer Chen, Yang Liu, Songjian Lu, Barry O’Sullivan, and Igor Razgon. 2008. A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55, 5. Google Scholar
Digital Library
- Marek Cygan, Stefan Kratsch, Marcin Pilipczuk, Michal Pilipczuk, and Magnus Wahlström. 2012. Clique cover and graph separation: New incompressibility results. In Proceedings of 39th ICALP. 254--265. Google Scholar
Digital Library
- Holger Dell and Dániel Marx. 2012. Kernelization of packing problems. In Proceedings of 23rd SIAM SODA. 68--81. Google Scholar
Digital Library
- Holger Dell and Dieter van Melkebeek. 2010. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In Proceedings of 42nd STOC. 251--260. Google Scholar
Digital Library
- Reinhard Diestel. 2010. Graph Theory 4th Ed. Springer-Verlag, Heidelberg.Google Scholar
- Michael J. Dinneen. 1997. Too many minor order obstructions. J. Univers. Comput. Sci. 3, 1199--1206.Google Scholar
- Michael J. Dinneen, Kevin Cattell, and Michael R. Fellows. 2001. Forbidden minors to graphs with small feedback sets. Discrete Math. 230, 1--3, 215--252.Google Scholar
Cross Ref
- Michael J. Dinneen and Rongwei Lai. 2007. Properties of vertex cover obstructions. Discrete Math. 307, 21, 2484--2500 Google Scholar
Digital Library
- Michael J. Dinneen and Liu Xiong. 2002. Minor-order obstructions for the graphs of vertex cover 6. J. Graph Theory 41, 3, 163--178. Google Scholar
Digital Library
- Rod Downey and Michael R. Fellows. 1999. Parameterized Complexity. Springer, New York. Google Scholar
Digital Library
- Andrew Drucker. 2012. New limits to classical and quantum instance compression. In Proceedings of 53rd FOCS. 609--618 Google Scholar
Digital Library
- Zdenek Dvorak, Archontia C. Giannopoulou, and Dimitrios M. Thilikos. 2012. Forbidden graphs for tree-depth. Euro. J. Comb. 33, 5, 969--979. Google Scholar
Digital Library
- Paul Erdős and Tibor Gallai. 1961. On the minimal number of vertices representing the edges of a graph. Publ. Math. Inst. Hung. Acad. Sci., Ser. A 6, 181--203.Google Scholar
- Michael R. Fellows, Bart M. P. Jansen, and Frances Rosamond. 2013. Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity. Euro. J. Combin. 34, 3, 541--566. Google Scholar
Digital Library
- Michael R. Fellows and Michael A. Langston. 1988. Nonconstructive tools for proving polynomial-time decidability. J. ACM 35, 3, 727--739. Google Scholar
Digital Library
- Michael R. Fellows and Michael A. Langston. 1989. An analogue of the Myhill-Nerode Theorem and its use in computing finite-basis characterizations (extended abstract). In Proceedings of 30th FOCS. 520--525. Google Scholar
Digital Library
- Jörg Flum and Martin Grohe. 2006. Parameterized Complexity Theory. Springer-Verlag, New York. Google Scholar
Digital Library
- Fedor V. Fomin, Bart M. P. Jansen, and Michal Pilipczuk. 2014. Preprocessing subgraph and minor problems: When does a small vertex cover help? J. Comput. Syst. Sci. 80, 2, 468--495. Google Scholar
Digital Library
- Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. 2012. Planar F-deletion: Approximation, kernelization and optimal FPT algorithms. In Proceedings of 53rd FOCS. 470--479. Google Scholar
Digital Library
- Lance Fortnow and Rahul Santhanam. 2011. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77, 1, 91--106. Google Scholar
Digital Library
- Michael R. Garey and David S. Johnson. 1979. Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York. Google Scholar
Digital Library
- Jiong Guo and Rolf Niedermeier. 2007. Invitation to data reduction and problem kernelization. SIGACT News 38, 1, 31--45. Google Scholar
Digital Library
- Danny Harnik and Moni Naor. 2010. On the compressibility of NP instances and cryptographic applications. SIAM J. Comput. 39, 5, 1667--1713. Google Scholar
Digital Library
- Danny Hermelin and Xi Wu. 2012. Weak compositions and their applications to polynomial lower bounds for kernelization. In Proceedings of 23rd SODA. 104--113. Google Scholar
Digital Library
- Bart M. P. Jansen. 2013. On sparsification for computing treewidth. In Proceedings of 8th IPEC. 216--229.Google Scholar
Cross Ref
- Bart M. P. Jansen and Stefan Kratsch. 2013. Data reduction for graph coloring problems. Inform. Comput. 231, 0, 70--88. Google Scholar
Digital Library
- Nancy G. Kinnersley. 1992. The vertex separation number of a graph equals its path-width. Inf. Process. Lett. 42, 6, 345--350. Google Scholar
Digital Library
- Stefan Kratsch. 2012. Co-nondeterminism in compositions: A kernelization lower bound for a Ramsey-type problem. In Proceedings of 23rd SODA. 114--122. Google Scholar
Digital Library
- Stefan Kratsch. 2014. Co-nondeterminism in compositions: A kernelization lower bound for a Ramsey-type problem. ACM Trans. Algor. 1--13. To appear. Google Scholar
Digital Library
- Stefan Kratsch, Marcin Pilipczuk, Ashutosh Rai, and Venkatesh Raman. 2012. Kernel lower bounds using co-nondeterminism: Finding induced hereditary subgraphs. In Proceedings of 13th SWAT. 364--375. Google Scholar
Digital Library
- Stefan Kratsch and Magnus Wahlström. 2011. The AND-conjecture may be necessary. In Parameterized Complexity Newsletter. FPT Wiki, Darwin.Google Scholar
- Stefan Kratsch and Magnus Wahlström. 2012. Representative sets and irrelevant vertices: New tools for kernelization. In Proceedings of 53rd FOCS. 450--459. Google Scholar
Digital Library
- Jens Lagergren. 1998. Upper bounds on the size of obstructions and intertwines. J. Comb. Theory, Ser. B 73, 1, 7--40. Google Scholar
Digital Library
- Choongbum Lee. 2009. On the size of minimal unsatisfiable formulas. Electr. J. Comb. 16, 1.Google Scholar
- Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. 2012. Kernelization - preprocessing with a guarantee. In The Multivariate Algorithmic Revolution and Beyond, 129--161.Google Scholar
- László Lovász and Michael D. Plummer. 1986. Matching Theory. Annals of Discrete Mathematics, Vol. 29. North-Holland Publishing Co., Amsterdam, The Netherlands.Google Scholar
- Neil Robertson and Paul D. Seymour. 1983. Graph minors. I. Excluding a forest. J. Comb. Theory, Ser. B 35, 1, 39--61.Google Scholar
- Neil Robertson and Paul D. Seymour. 1986. Graph minors. V. Excluding a planar graph. J. Comb. Theory, Ser. B 41, 1, 92--114. Google Scholar
Digital Library
- Neil Robertson and Paul D. Seymour. 1995. Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63, 1, 65--110. Google Scholar
Digital Library
- Neil Robertson and Paul D. Seymour. 2004. Graph Minors. XX. Wagner’s conjecture. J. Comb. Theory, Ser. B 92, 2, 325--357. Google Scholar
Digital Library
- Juanjo Rué, Konstantinos S. Stavropoulos, and Dimitrios M. Thilikos. 2012. Outerplanar obstructions for a feedback vertex set. Eur. J. Comb. 33, 5, 948--968. Google Scholar
Digital Library
- Atsushi Takahashi, Shuichi Ueno, and Yoji Kajitani. 1994. Minimal acyclic forbidden minors for the family of graphs with bounded path-width. Discrete Math. 127, 293--304. Google Scholar
Digital Library
- Chee-Keng Yap. 1983. Some consequences of non-uniform conditions on uniform classes. Theor. Comput. Sci. 26, 287--300.Google Scholar
Cross Ref
Index Terms
FPT is characterized by useful obstruction sets: Connecting algorithms, kernels, and quasi-orders
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