1. AGAPE Corsica 2009 Introduction and overview of FPT algorithmics Michael Fellows University of Newcastle, Australia

2. AGAPE Corsica 2009 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.” What is FPT?

3. AGAPE Corsica 2009 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. AGAPE Corsica 2009 The parameterized framework is two-dimensional n = input size k = a relevant secondary measure f(k)n c n g(k) vs “good” FPT “bad” W-hard, etc. Complexity frameworks are driven by contrasting function classes.

5. AGAPE Corsica 2009 Frameworks in pictures The classical P vs NP framework k ncnc n Intrinsic Combinatorial explosion: Most problems are NP-hard or worse. The parameterized framework FPT Try to confine the explosion to the parameter.

6. AGAPE Corsica 2009 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 A simple view: (problems of interest) + (reductions) = empirical complexity classes.

7. AGAPE Corsica 2009 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 M p ) ? Will it ever halt? Undecidable HP II in A Turing machine M nondeterministic. ? Can M halt in | M | steps? HP III in A nondeterministic Turing machine M (unlimited nondeterminism & alphabet size), k. ? Can M halt in < k steps? Trivially (by def.) complete for NP Complete for W[1]

8. AGAPE Corsica 2009 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 M p ) ? Will it ever halt? Undecidable HP II in A Turing machine M nondeterministic. ? Can M halt in | M | steps? HP III in A nondeterministic Turing machine M (unlimited nondeterminism & alphabet size), k. ? Can M halt in < k steps? Trivially (by def.) complete for NP Complete for W[1]

9. AGAPE Corsica 2009 Downey-Fellows: Parameterized Complexity, Springer, 1999 Flum-Grohe: Parameterized Complexity Theory, Springer, 2006. Niedermeier: Invitation to Fixed- Parameter Algorithms, Oxford University Press, 2006.

10. AGAPE Corsica 2009 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 volume 51, number 1 January 2008 the Computer journal

11. AGAPE Corsica 2009 volume 51, number 3 May 2008 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 Comparison Erik 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 Applications Gregory 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 theComputer journal

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13. AGAPE Corsica 2009 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. Two complementary mathematical toolkits:

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15. AGAPE Corsica 2009 for k = number of cities by “try all permutations” implies k! n FPT Shows that a parameterized problem can be “trivially FPT” But it is still interesting to look for “better FPT” Such as 2 k n Of course TSP is also in FPT

16. AGAPE Corsica 2009 (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 When a parameterized problem is shown FPT, two races begin

17. AGAPE Corsica 2009 SET SPLITTING: In: Family F  2 X 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  F if there exists a  S, a  X – X ’ and there exists a ’  S, a ’  X ’. Dehne et al (2003): O(72 k n 3 ) Dehne et al (2004): O*(8 k ) “Crown reduction” Lokshtanov and Sloper (2005): O*(2.7 k ) Chen, Liu (2008):O*(2 k ) “Randomized disposal” Another example: SET SPLITTING f(k) race

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20. AGAPE Corsica 2009 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( k 11 ) P-time kernelization Bodlaender et al. (2007) O( k 3 ) kernel Thomasse (2009) O( k 2 ) kernel Example of a “kernelization race”

21. AGAPE Corsica 2009 (1)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? 3 simple graph problems that drove the field NP-complete but P for any fixed k NP-complete but P for any fixed k NP-complete but P for any fixed k

22. AGAPE Corsica 2009 (1)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) What was found

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