1 / 30

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.”.

reina
Télécharger la présentation

Introduction and overview of FPT algorithmics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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)

More Related