introduction and overview of fpt algorithmics
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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.”. - PowerPoint PPT PresentationTRANSCRIPT
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Introduction and overview of Introduction and overview of FPT algorithmicsFPT algorithmics
Michael Fellows
University of Newcastle, Australia
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• 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?
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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.
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The parameterized framework is two-dimensional
n = input sizek = a relevant secondary measure
f(k)nc n g(k)vs
“good”
FPT
“bad”
W-hard, etc.
Complexity frameworks are driven by contrasting function classes.
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Frameworks in pictures
The classical P vs NP framework
k
nc
nIntrinsic Combinatorial explosion: Most problems are NP-hard or worse.
The parameterized framework
FPT
Try to confine the explosion to the parameter.
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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.
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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 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]
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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 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]
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Downey-Fellows: Parameterized Complexity, Springer, 1999
Flum-Grohe: Parameterized Complexity Theory, Springer, 2006.
Niedermeier: Invitation to Fixed-Parameter Algorithms, Oxford University Press, 2006.
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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
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volume 51, number 3 May 2008
Hans L. Bodlaender and Arie M. C. A. Koster Combinatorial Optimization on Graphs of Bounded TreewidthLiming 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 GottlobWidth 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
the Computer journal
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• 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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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 2kn
Of course TSP is also in FPT
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(1) The “f(k) race”: The race to findBetter 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
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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 F if there exists a S, a X – X’ and there exists a’ S, a’ 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”
Another example: SET SPLITTING
f(k) r
ace
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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
Example of a “kernelization race”
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(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 fieldNP-complete
but P for any fixed k
NP-complete
but P for any fixed k
NP-complete
but P for any fixed k
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(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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