1. Fixed Parameter Tractability
Ron Hagan

2. Prehistory
Neil Robertson, Paul Seymour published a series of papers in the Journal of Combinatorial Theory Series B.
Beginning with Graph Minors.I.Excluding a Forest, appearing and and currently up to Graph Minors.XXIII.Nash-Williams’ Immersion Conjecture. The most recent appearing in 2012.
One of the main intended results culminated in Graph Minors.XX.Wagner’s Conjecture, in a proof of what is now known as The Graph Minor Theorem.

3. Prehistory
In 1985 Mike Fellows was hired at Washington State University where he and Mike Langston began exploring the practical implications of Robertson and Seymour’s work and well quasi orderings.

4. Prehistory
If a family of graphs is closed under taking minors, then membership in that family can be tested in polynomial time.
Problems:
1) The algorithm is non-constructive. (requires knowledge of obstruction set)
2) It hides huuuuuuuuge constants of proportionality.

5. Prehistory
Dr. Langston and Mike Fellows pioneering work in applications included proofs that:
For every fixed k, gate matrix layout is solvable in polynomial time.
As well as analogs for:
Disk dimension
Minimum cut linear arrangement
Topological bandwidth
Crossing number
Maximum leaf spanning tree
Search number
Two dimensional grid load factor

6. Prehistory
Their work would lay the foundation for what would be formalized as a new field of study – fixed parameter tractability.
R.G. Downey and M.R. Fellows. Parameterized Complexity. Springer-Verlag 1999.

7. Other Early Players
J. Flum and M. Grohe: Parameterized Complexity Theory, Springer, 2006
R. Niedermeier: Invitation to Fixed-Parameter Algorithms, Oxford University Press, 2006

8. Motivation Classical complexity theory is very coarsely grained.
Focuses on worst case behavior.
Only takes into account the size of the problem.

9. Motivation Parametric complexity adopts a more fine grained approach.
Seeks to acknowledge and make use of aspects in the structure of the inputs that may affect complexity.
FPT algorithms seek to confine the combinatorial explosion to some parameter independent of the input size.

10. Motivation All NP-complete problems are not created equally.
Dominating set, still 𝑂( 𝑛 𝑘+1 ) by brute force.
Vertex cover 𝑂( 2 𝑘 𝑛) via simple search tree.
Current best is 𝑂( 𝑘 +𝑘𝑛) due to Chen, Kanj, and Xia.

11. Parametric Complexity
A parameterized problem is a language , where is a finite alphabet. The second component is called the parameter of the problem.
A parameterized problem is fixed-parameter tractable if membership can be determined in time, where f is a computable function depending only on k. The corresponding complexity class is called FPT.

12. Parametric Complexity
The class of problems which are fixed-parameter tractable is denoted by FTP.
Vertex cover – trivially FPT by brute force.
Dominating Set, Clique, and Independent set are not FPT (when parameterized by the size of the solution).

13. Parametric Complexity – Reductions
Definition: Let be two parameterized problems. is said to reduce to by a standard parameterized reduction if there are functions and from ℕ to ℕ and a function from to such that:
is computable in time for some constant c, and
if and only if

14. Parametric Complexity – The W-Hierarchy
A Boolean formula is in conjunctive normal from CNF if it is written as a product of sums – an AND of ORs ( 𝑥 1 ∨𝑥 2 )∧( 𝑥 2 ∨𝑥 3 ∨𝑥 4 ) An assignment of variables can be weighted by the number of variables set to true. The example above has a minimum weight 1 satisfying assignment.

15. Parametric Complexity –The W-Hierarchy
WEIGHTED CNF-SATISFIABILITY Input: A Boolean formula F in CNF and a nonnegative integer k. Question: Is there a satisfying assignment with weight exactly k? The Weighted 2-CNF-SAT problem is not known to be FPT – if it is, it implies the collapse of the W-hierarchy.

16. Parametric Complexity –The W-Hierarchy
Definition: A boolean formula is t-normalized if it can be written in the form of a “product-of-sums-of-products-…” of literals with t-1 alternations between products and sums.
2-CNF formulas are 1-normalized
CNF formulas are 2-normalized
t is the weft of the associated boolean circuit

17. Parametric Complexity –The W-Hierarchy
𝑊 𝑡 for 𝑡≥1 is the class of all parameterized problems that can be reduced to WEIGHTED t-NORMALIZED SATISFIABILITY by a parameterized reduction.
𝑊[𝑆𝐴𝑇] is the class of parameterized problems that are fpt reduceable to weighted satisfiability for boolean formulas.
𝑊[𝑃] is the class of parameterized problems that are fpt reduceable to weighted satisfiability for boolean circuits.
𝑊 𝑋𝑃 is the class of parameterized problems that are solvable in 𝑓(𝑘)⋅ 𝑥 𝑔(𝑘) time.
𝐹𝑃𝑇⊆𝑊 1 ⊆𝑊 2 ⊆…𝑊 𝑡 ⊆…𝑊[𝑆𝐴𝑇]⊆𝑊[𝑋𝑃]

18. Parametric Complexity
Hardness and completeness are defined analogously to classical complexity. Theorem: WEIGHTED 2-CNF-SATISFIABILITY is fpt-reduceable to WEIGHTED ANTIMONOTONE 2-CNF-SATISFIABILITY. So WEIGHTED ANTIMONOTONE 2-CNF-SATISFIABILITY is W[1]- complete.

19. Parametric Complexity - Examples
A graph G=(V,E) with V={1,2,…,n} has an independent set of size k if and only if the antimonotone 2-CNF formula has a weight-k satisfying truth assignment. It follows that Independent Set is W[1]-complete.

20. Parametric Complexity - Examples
A graph G=(V,E) with V={1,2,…,n} has a dominating set of size k if and only if the CNF formula has a weight-k satisfying truth assignment. It follows that Dominating Set is in W[2]. It is actually W[2]-complete.

21. Fixed-Parameter Tractability
From the perspective of algorithm design, there are two avenues to attack a problem that has been shown to be FPT.
Reduce the cost of the dependence on k.
Reduce the cost of the remaining dependence on the size of the problem.
This can be approached many ways, but two of the main techniques are through kernelization and bounded search trees.

22. Kernelization
A kernelization for a parameterized problem L is an algorithm that maps in polynomial time an instance x with parameter k to an instance x’ with parameter k’ such that
x is in L if and only if x’ is in L and
k’ and the size of x’ are both bounded by computable functions of k.
It can be shown that being FPT is equivalent to admitting a kernel.

23. Kernelization – Vertex Cover
Reduction Rules
Remove all isolated vertices.
Degree 1 rule: Remove all degree one vertices, add their neighbors to the solution set and remove them and all their incident edges from the graph, reduce k by one for each vertex added to the solution set.
High degree rule: If a vertex has degree greater than k, add it to the solution set, remove it from the graph, and reduce k by one.
Exhaustively applying these rules leads to a kernel of size at most 𝑘 2 (first observed by Sam Buss). If more than 𝑘 2 vertices remain, the graph can’t have a cover of size 𝑘.

24. Branching – Vertex Cover
Branching by either including a node or including all of it’s neighbors gives a 𝑂( 5 𝑘 4 ⋅𝑛) algorithm.
Niedermeier used a set of simple branching rules to bound the search tree above by 𝑂( 1.47 𝑘 ).
More involved rules for exploring the local structure in neighborhoods produce the running time of 𝑂( 𝑘 ) for the branching in the algorithm of Chen et. al.

25. Vertex Cover
Combining the kernelization and branching strategies we can produce an algorithm for vertex cover which runs in 𝑂( 𝑘 ) 𝑘 2 time. The performance can be further improved through techniques such as interleaving by rekernelizing children at set levels in the search tree.

26. Current Research
FPT algorithms and parameterized complexity are an active area of research. A webpage is maintained at containing everything from the current state of fpt races to a list of open problems.

27. References
Langston, Michael A. "Fixed-parameter tractability, a prehistory." The Multivariate Algorithmic Revolution and Beyond. Springer Berlin Heidelberg,
Niedermeier, Rolf. "Invitation to fixed-parameter algorithms, volume 31 of Oxford Lecture Series in Mathematics and its Applications." (2006).
Downey, Rodney G., and Michael Ralph Fellows. Parameterized complexity. Springer Science & Business Media, 2012.

28. References
J. Flum, M. Grohe, Parameterized Complexity Theory, Springer-Verlag,
Chen, Jianer, Iyad A. Kanj, and Ge Xia. "Improved upper bounds for vertex cover." Theoretical Computer Science  (2010):
Buss, Jonathan F., and Judy Goldsmith. "Nondeterminism within P^*." SIAM Journal on Computing 22.3 (1993):

29. Homework
Prove that maximum clique parameterized by the size of the solution is W[1]-complete.
Recall that maximum clique can be solved by finding the minimum vertex cover in the complement of a graph. Explain clearly and succinctly why this fact is not in conflict with vertex cover being FPT.
Let G = (V,E) be a simple connected graph with |V|>1. Prove that if G has treewidth 1, the vertex cover problem parameterized by k = the size of the cover is solvable by kernelization alone for any k.