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Introduction and overview of FPT algorithmics

Introduction and overview of FPT algorithmics. Michael Fellows University of Newcastle, Australia. What is FPT?. A quick overview of parameterized complexity “A two-dimensional sequel to P vs NP and all that.” “An opening chapter of multivariate complexity analysis and algorithm design.”.

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Introduction and overview of FPT algorithmics

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  1. Introduction and overview of FPT algorithmics Michael Fellows University of Newcastle, Australia

  2. What is FPT? • A quick overview of parameterized complexity • “A two-dimensional sequel to P vs NP and all that.” • “An opening chapter of multivariate complexity analysis and algorithm design.”

  3. The “classical” P vs NP framework is one-dimensional n = input size poly(n) 2 poly(n) vs “good” P positive toolkit of how to design P-time algorithms “bad” NP, etc. negative toolkit of NP-hardness, etc. Unfortunately, almost everything turns out to be NP-hard.

  4. The parameterized framework is two-dimensional n = input size k = a relevant secondary measure vs f(k)nc n g(k) “good” FPT “bad” W-hard, etc. Complexity frameworks are driven by contrasting function classes.

  5. Frameworks in pictures The classical P vs NP framework Intrinsic Combinatorial explosion:Most problems are NP-hard or worse. n The parameterized framework k FPT nc Try to confine the explosion to the parameter.

  6. A simple view: (problems of interest) + (reductions) = empirical complexity classes. The main parameterized hierarchy: current P  lin(k)  poly(k)  FPT  M[1]  W[1]  M[2]  W[2] ... W[SAT]  W[P] ... XP P The best kind of FPT is P kernelization classes The analog of NP Intractable

  7. W[1] is the natural two-dimensional analog of NP Our premier guides to intractability are various forms of the HALTING PROBLEM. • HP I • in A program P (Turing machine Mp) • ? Will it ever halt? Undecidable • HP II • in A Turing machine Mnondeterministic. • ? Can M halt in |M | steps? Trivially (by def.) complete for NP • HP III • in A nondeterministic Turing machine M (unlimited nondeterminism & alphabet size), k. • ? Can M halt in <k steps? Complete for W[1]

  8. W[1] is the natural two-dimensional analog of NP Our premier guides to intractability are various forms of the HALTING PROBLEM. • HP I • in A program P (Turing machine Mp) • ? Will it ever halt? Undecidable • HP II • in A Turing machine Mnondeterministic. • ? Can M halt in |M | steps? Trivially (by def.) complete for NP • HP III • in A nondeterministic Turing machine M (unlimited nondeterminism & alphabet size), k. • ? Can M halt in <k steps? Complete for W[1]

  9. Downey-Fellows: Parameterized Complexity, Springer, 1999 Flum-Grohe: Parameterized Complexity Theory, Springer, 2006. Niedermeier: Invitation to Fixed-Parameter Algorithms, Oxford University Press, 2006.

  10. volume 51, number 1 January 2008 the Computerjournal • Rodney G. Downey, Michael R. Fellows, and Michael A. Langston • Foreword by the Guest Editors Falk Hüffner, Rolf Niedermeier, and Sebastian Wernicke • Techniques for Practical Fixed-Parameter Algorithms Michael A. Langston, Andy D. Perkins, Arnold M. Saxton, • Jon A. Scharff, and Brynn H. Voy • Innovative Computational Methods for Transcriptomic Data Analysis: A Case Study in the Use of FPT for Practical Algorithm Design and Implementation Jianer Chen and Jie Meng • On Parameterized Intractability: Hardness and Completeness Dániel Marx • Parameterized Complexity and Approximation Algorithms Jens Gramm, Arfst Nickelsen, and Till Tantau • Fixed-Parameter Algorithms in Phylogenetics • Leizhen Cai • Parameterized Complexity of Cardinality Constrained Optimization Problems • Christian Sloper and Jan Arne Telle • An Overview of Techniques for Designing Parameterized Algorithms • Book review • William Gasarch and Keung Ma Kin • Invitation to Fixed-Parameter Algorithms • Parameterized Complexity Theory • Parameterized Algorithmics: Theory, Practice and Prospects

  11. volume 51, number 3 May 2008 the Computerjournal • Hans L. Bodlaender and Arie M. C. A. Koster • Combinatorial Optimization on Graphs of Bounded Treewidth • Liming Cai, Xiuzhen Huang, Chunmei Liu, Frances Rosamond, • and Yinglei Song • Parameterized Complexity and Biopolymer • Sequence ComparisonErik D. Demaine and MohammadTaghi Hajiaghayi • The Bidimensionality Theory and Its Algorithmic Applications Georg Gottlob and Stefan Szeider • Fixed-Parameter Algorithms For Artificial Intelligence, Constraint Satisfaction and Database Problems • Petr Hlineny, Sang-il Oum, Detlef Seese, and Georg Gottlob • Width Parameters Beyond Tree-width and their ApplicationsGregory Gutin and Anders Yeo • Some Parameterized Problems On Digraphs • Panos Giannopoulos, Christian Knauer, and Sue Whitesides • Parameterized Complexity of Geometric Problems • Iris van Rooij and Todd Wareham • Parameterized Complexity in Cognitive Modeling: Foundations, Applications and Opportunities

  12. Two complementary mathematical toolkits: • How to design efficient algorithms The positive toolkit of FPT algorithm techniques. • How to analyze complexity and recognize intractability. The negative toolkit of M[1] and W[1] hardness.

  13. Of course TSP is also in FPT for k = number of cities by “try all permutations” impliesk! nFPT Shows that a parameterized problem can be “trivially FPT” But it is still interesting to look for “better FPT” Such as 2kn

  14. When a parameterized problem is shown FPT, two races begin (1) The “f(k) race”: The race to find Better and better (slower growing) f(k) (2) The “kernelization race”: The race to find Smaller and smaller P-time kernelizations

  15. Another example: SET SPLITTING SET SPLITTING: In: Family F 2X of subsets of a base set X. Parameter: k Question: Does there exist X’ X that that splits at least k sets in F ? Where X’ splits S e F if there exists a e S, a e X – X’ and there exists a’ e S, a’ e X’. Dehne et al (2003): O(72k n3) Dehne et al (2004): O*(8k) “Crown reduction” Lokshtanov and Sloper (2005): O*(2.7k) Chen, Liu (2008):O*(2k) “Randomized disposal” f(k) race

  16. Example of a “kernelization race” UNDIRECTED FEELBACK VERTEX SET: In: (G, k) Parameter: k Question: Is it possible to delete at most k vertices from G to get a graph G’ that is acyclic? • Known to be FPT for many years, but no poly(k) kernelization • Burrage et al. (2006) O(k11) P-time kernelization • Bodlaender et al. (2007) O(k3) kernel • Thomasse (2009) O(k2) kernel

  17. 3 simple graph problems that drove the field NP-complete but P for any fixed k • VERTEX COVER In: (G, k) Parameter: k Question: Are there k vertices that cover all the edges? (2) CLIQUE In: (G, k) Parameter: k Question: Does G have a k clique? (3) DOMINATING SET In: (G, k) Parameter: k Question: Are there k vertices that cover all the vertices of G? NP-complete but P for any fixed k NP-complete but P for any fixed k

  18. What was found • VERTEX COVER is linear-time for any fixed k by “bounded search tree” approach (2) Both k-CLIQUE and k-DOM SET seem to require something more like brute force: “try all k-subsets” naively O(n k+1)

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