1. Parameterised Complexity
Algorithms and Networks 2016/2017
Johan M. M. van Rooij
Hans L. Bodlaender

2. This lecture Classical complexity: Parameterised complexity:
P vs NP.
Conditional non-existence proofs.
Parameterised complexity:
FTP.
XP.
The W-hierarchy.
Clique is not in FPT.
Parameterised reductions / hardness proofs.
Clique on regular graphs.
Multicoloured clique on regular graphs.
Dominating set.

3. Classical complexity and conditional proofs
Parameterised Complexity – Algorithms and Networks
Classical complexity and conditional proofs

4. Classical complexity: P
Classical complexity theory defines a problem to be ‘tractable’ if it can be solved by a polynomial-time algorithm.
P is the class of polynomial time solvable problems.
Shortest paths.
Maximum matching / minimum weight perfect matching.
Maximum flow / minimum cost flow.
Stable marriage.
Many more.
Classical complexity theory defines a problem to be ‘intractable’ if it cannot be solved by a no polynomial-time algorithm.
Even so, we often solve many problems that are not in P in practical situations (most interesting problems are not in P).

5. Classical complexity: NP
NP is the class of all problems for which we can verify solutions in polynomial time.
We have seen before: P µ NP.
NP includes many problems that we have considered so far:
All problems in P.
Graph isomorphism.
Satisfiability / 3-Satisfiability / Max-Satisfiability.
Maximum Independent Set / Minimum Vertex Cover / Clique.
Minimum Dominating Set.
Set Cover.
Hamilton Cycle / Travelling Salesman Problem.
Note that the list contains problems in P, problems that are NP-complete, and problems (graph isomorphism) for which we do not know whether they are in P or whether they are NP-complete.

6. Classical complexity: P vs NP
We have not been able to prove that P = NP or P  NP.
Therefore, we do not know whether there are problems in NP without polynomial-time algorithms.
But we expect that P  NP, and thus that some problem in NP do not have polynomial-time algorithms.
We have been able to prove that if some problems have polynomial-time algorithms solving them, then all problems in NP have polynomial-time algorithms.
These problems are called NP-complete.
Cook-Levin Theorem: Satisfiability is NP-complete.
Polynomial-time reductions prove other problems to be NP-complete as well.
Question: what means NP-hard compared to NP-complete here?

7. Conditional non-existence proofs
Proving a problem to be NP-complete.
Does not prove that no polynomial-time algorithm exists for the problem.
Does prove that: if P  NP, then no polynomial-time algorithm exists for the problem.
This is a conditional non-existence proof.
If a complexity-theoretic assumption holds, then some algorithm cannot exist.
We have seen some examples when considering approximation algorithms: if P  NP, then some approximation algorithms cannot exist.
Conditional non-existence proofs do not prove that some type of algorithm does not exist.
Only that it has huge scientific consequences if one would exists.
And very likely requires deep new insights in complexity theory.
Still, do not fall for the ‘I solved P=?NP question trap’.

8. Parameterised complexity
Parameterised Complexity – Algorithms and Networks
Parameterised complexity

9. Parameterised complexity
In parameterised complexity, we consider parameterised decision problems (different from decision problems in NP):
Instance (often with size denoted as n).
Parameter (often a number k, but can be something else).
Yes/No decision question.
FPT: class of parameterised problems that can be solved in O( f(k) poly(n) ) time.
What if P = NP?
What about FPT then?
Can you give an example of something else?

10. Parameterised complexity
Parameterised complexity classes:
FPT.
W[1] µ W[2] µ ....
XP.
FPT µ W[1] µ W[2] µ ... µ XP.
FPT we know.
XP: class of parameterised problems that can be solved in O( nf(k) ) time.
W[i] classes we will not formally define (see book on Parameterised Complexity if you are interested).
However, we will use that some problems are W[1]-hard.
We will get back to this – we need some theory first.
XP: Can you give some examples of problems in XP? (anything you know to be FTP, dominating set, independent set, clique, etc...)
XP useful in thinking about parameterised problems – hard to find FTP algorithm if an XP algorithm is not easily found.

11. Parameterised reductions
In classical complexity polynomial-time reductions from problem A to problem B prove that:
If B is polynomial-time solvable, then A is.
But also, if A is most-likely not in P, then so is B most likely.
In parameterised complexity, we have parameterised reductions from problem A to problem B.
If B is in FPT, then so is A.
And: if A is not in FPT (under some assumption that we get to later), then B is not either.
Note the change of order of A and B here!

12. Parameterised reduction: definition
A parameterised reduction from parameterised problem A to parameterised problem B is an algorithm R that given an instance and parameter (x,k) of A outputs and instance and parameter (x’,k’) of B such that:
(x,k) is a yes-instance, if and only if, (x’,k’) is a yes instance.
k‘ · g(k) for some computable function g.
R runs in f(k)poly(x) time, for some computable function f.
Lemma (parameterised reductions work as intended):
If B is in FPT and there is a parameterised reduction from A to B, then A is in FPT.
Proof on blackboard, good exercise to do yourself.
Use that if you run a reduction R on an instance (x,k) of A, then the size of the result is at most the running time of R.

13. Parameterised clique problem
Instance: graph G=(V,E).
Parameter: integer k.
Question: does G have a clique of size at least k?
In parameterised complexity, we often work with the assumption that clique is not in FPT.
This assumption is equivalent to: FPT  W[1].
This is true because clique is W[1]-complete.
I want to omit the (rather technical) definition of the W[i] classes. FPT µ W[1] µ W[2] µ ... µ XP.
I do want you to know that these two statements are equivalent.

14. Parameterised complexity and conditional non-existence proofs
So FPT  W[1] is equivalent to clique not being in FPT.
FPT  W[1] is an often used assumption in parameterised complexity.
How does this relate to P  NP? Is it stronger? It is weaker?
Independent Set
Instance: graph G=(V,E).
Parameter: integer k.
Question: does G have an independent set of size at least k?
Assuming FPT  W[1], what can we say about the parameterised version of independent set?
Clique in G is equivalent to an independent set in the complement of G.

15. Independent set
Assuming FPT  W[1], what can we say about the parameterised version of independent set?
Let us prove that assuming FPT  W[1], independent set is not in FPT by giving a parameterised reduction.
Recall:
A clique in G is equivalent to an independent set in the complement of G.
A parameterised reduction from problem A to problem B is an algorithm R that given an instance and parameter (x,k) of A outputs and instance and parameter (x’,k’) of B such that:
(x,k) is a yes-instance, if and only if, (x’,k’) is a yes instance.
k‘ · g(k) for some computable function g.
R runs in f(k)poly(x) time, for some computable function f.
(Very simple) reduction R:
Take the complement of the graph G, do not change k.

16. Vertex cover Assuming FPT  W[1], What about vertex cover?
Clique is not in FPT.
Independent set is not in FPT.
What about vertex cover?
I is an independent set in G=(V,E), iff, V-I is a vertex cover in G.
Vertex Cover
Instance: graph G=(V,E).
Parameter: integer k.
Question: does G have an vertex cover of size at most k?
Hans showed you already that vertex cover is FPT.
Why does a similarly simple reduction not work?
Because the parameter changes! Parameterised reductions and polynomial-time reductions are mostly different!

17. More Parameterised REDUCTIONS
Parameterised Complexity – Algorithms and Networks
More Parameterised REDUCTIONS

18. More parameterised reductions
A parameterised reduction from parameterised problem A to parameterised problem B is an algorithm R that given an instance and parameter (x,k) of A outputs and instance and parameter (x’,k’) of B such that:
(x,k) is a yes-instance, if and only if, (x’,k’) is a yes instance.
k‘ · g(k) for some computable function g.
R runs in f(k)poly(x) time, for some computable function f.
Next reduction will be to clique on regular graphs.
Clique on regular graphs
Instance: graph G=(V,E) where every vertex has degree d.
Parameter: integer k.
Question: does G have a clique of size at least k?

19. A parameterised reduction to clique on regular graphs
Parameterised reduction: clique to clique on regular graphs.
Let G=(V,E) be the instance of clique with parameter k.
If k · 2, clique is trivial and we can output a trivial solution.
Build an instance of clique on regular graphs as follows:
Let d be the maximum degree in G (example below has d=4).
Take d disjoint copies G1, G2, ..., Gd. Let vi be the copy of v in Gi.
For every v2V, add d - degree(v) vertices, and add edge between these vertices and all copies vi.
Set the new parameter k’ equal to k. G2 G3 G4 G1

20. Two (or five) things to prove: this is a parameterised reduction
The result is a d-regular graph
That is: every vertex has degree d.
True for vertices in Gi because of the extra edges. True for the new vertices as they have degree d by construction.
The construction is a parameterised reduction.
(G,k) is a yes-instance, if and only if, (G’,k’) is a yes instance.
If G has a clique of size k, clearly G’ has as it contains copies of G.
If G’ has a clique of size k ¸ 3, it cannot involve the new vertices as they are not contained in triangles, thus G must have it also.
k‘ · g(k) for some computable function g.
k‘ = k.
R runs in f(k)poly(|G|) time, for some computable function f.
R requires polynomial time.
So what about independent set on regular graphs?

21. Clique on regular graphs: conclusion
We have a parameterised reduction from clique to clique on regular graphs.
Thus, assuming FPT  W[1], clique on regular graphs in not in FPT (as so for clique).
Next we can use clique on regular graphs to reduce from.
Using that: if B is in FPT and there is a parameterised reduction from A to B, then A is in FPT.
I.e.: assuming that A is not in FPT, then B is not in FPT either.
What about independent set on regular graphs?

22. From clique on regular graphs to multicoloured clique on regular graphs
Instance: graph G=(V,E) where every vertex has degree d, and a colouring of the vertices in k colour classes V1,V2,..,Vk.
Parameter: integer k.
Question: does G have a clique of size k where each vertex has a different colour?
Parameterised reduction: from clique on regular graphs.
Let G be the d-regular input graph, and k the input parameter.
Construct an instance as follows:
For every vertex v in G, create k vertices v1,v2,..,vk, one of each colour.
Add edges between vertices vi and uj, if and only if, ij and u and v are adjacent in G.
Set the new parameter k’ equal to k.

23. Proving correctness of the reduction
Parameterised reduction: from clique on regular graphs.
Let G be the d-regular input graph, and k the input parameter.
Construct an instance as follows:
For every vertex v in G, create k vertices v1,v2,..,vk, one of each colour.
Add edges between vertices vi and uj, if and only if, ij and u and v are adjacent in G.
Set the new parameter k’ equal to k.
The result is a (k-1)d-regular graph.
So the result is an instance of multicoloured clique on regular graphs.
G has a k-clique, if and only if, the resulting graph has a k-clique using k different colours.

24. Multicoloured clique: conclusion
It indeed is a parameterised reduction.
G has a k-clique, if and only if, the resulting graph has a k-clique using k different colours.
k‘ = k (thus k’ · g(k) for some computable function g)
The reduction uses polynomial time (thus f(k)poly(n) time).
Thus, assuming FPT  W[1], multicoloured clique on regular graphs is not in FPT.
Also, assuming FPT  W[1], multicoloured clique is not in FPT.
Why?
And, multicoloured independent set?
Where every vertex in the independent set must have a different colour.

25. Last reduction: multicoloured independent set to dominating set
Given an instance of multicoloured independent set G=(V,E) with colour classes V1,V2,...,Vk, with parameter k.
Construct an instance of dominating set as follows:
Create a vertex for every vertex in G, and add edges so that every colour class Vi becomes a clique.
For every colour class Vi add two new vertices xi, yi adjacent to all vertices in Vi but not to each other.
For every edge {u,v}2E, with u2Vi, v2Vj add a vertex w{u,v} adjacent to all vertices in Vi and Vj except u and v.
Set the new parameter k’ to k. V1 V2 x1 x2 v
W{u,v} u y2 y1

26. Last reduction: intuition
The intuition of the reduction:
The vertices xi and yi ensure that a dominating set of size k in the constructed graph must take one vertex from each set Vi.
The vertices w{u,v} ensure that the selected vertices form an independent set in the original graph G: if both sides of an edge in {u,v} are selected, then w{u,v} is not dominated.
Formal proof that this is a parameterised reduction is not so difficult using the above intuition. V1 V2 x1 x2 v
W{u,v} u y2 y1

27. Parameterised Complexity – Algorithms and Networks
Conclusion

28. Concluding remarks
We have shown that, assuming FPT  W[1], the following problems are not in FPT:
Independent set.
Clique on regular graphs.
Multicoloured clique.
Multicoloured clique on regular graphs.
Multicoloured independent set.
Multicoloured independent set on regular graphs.
Dominating Set.
A good exercise to do yourself.
Prove that, FPT  W[1], there set cover (with the number of sets to choose as parameter) is not in FPT.

29. Concluding remarks: W[1] and W[2]
Stacking the reductions gives a parameterised reduction from independent set to dominating set.
You will not find a parameterised reduction from dominating set back to independent set.
This is different from polynomial-time reductions, where you can make a reduction between any two NP-complete problems.
The reason is that:
Independent set is W[1]-complete.
Dominating set is W[2]-complete.
FPT µ W[1] µ W[2] µ ... µ XP.
I don’t want you to understand the details of the W-hierarchy, just remember this as background information.