1. Joint work with Hans Bodlaender
Bart Jansen Independent Set Kernelization for a Refined Parameter: Upper and Lower bounds
Joint work with Hans Bodlaender
Algorithm seminar, Bergen
January 7th, 2011

2. Independent Set Kernelization for a Refined Parameter: Upper and Lower bounds
Introduction
Independent Set
Parameters
Kernelization
Upper bounds
Small kernel for parameter P3 cover
Reduction rules
Analysis
Ideas for for parameter Feedback Vertex Set
Lower bounds
Effect of introducing vertex weights
Conclusion

3. Our target problem
independent set

4. Independent Set Input: Graph G, integer q
Question: Is there a set S of ≥ q vertices which are pairwise non-adjacent?
NP-complete, even on planar graphs max degree 3
Not approximable
We show how to attack the problem if some measure of “graph complexity” is low
Data reduction

5. Solutions to vertex deletion problems as complexity measures
Parameters

6. Vertex Deletion Problems
Vertex Cover
Input: Graph G, integer q
Question: Is there a set S of ≤ q vertices such that G-S is edgeless?
Equivalent question:
Is there an Independent Set of size ≥ n – q?
Vertex Cover
Edgeless Graphs

7. Vertex Deletion Problems
P3 Cover
Input: Graph G, integer q
Question: Is there a set S of ≤ q vertices such that G-S is a collection of paths on at most 2 vertices?
P3 cover
Paths ≤2 nodes
Vertex Cover
Edgeless Graphs

8. Vertex Deletion Problems
Feedback Vertex Set
Input: Graph G, integer q
Question: Is there a set S of ≤ q vertices such that G-S is a forest? (Acyclic)
Feedback vtx Set
Forests
P3 cover
Paths ≤2 nodes
Vertex Cover
Edgeless Graphs

9. Graph Complexity Measures
We can use the minimum sizes of these vertex deletion sets as measures of the complexity of a graph
Every edgeless graph is a collection of paths on ≤ 2 nodes
Every collection of paths on ≤ 2 nodes is a forest
Difference between the parameters can be unbounded
Feedback vtx Set
Forests
P3 cover
Paths ≤2 nodes
Vertex Cover
Edgeless Graphs ≤ ≤

10. Collections of paths on ≤ 2 vertices
Graph families
All graphs
Forests
Collections of paths on ≤ 2 vertices
Edgeless graphs

11. Attacking hard problems with small parameters
kernelization

12. Graph problems with structural parameters
Consider a computational decision problem on graphs
Input: encoding x of a question about graph G, integer k.
Question: does graph G have a (…)?
Parameter:k
Parameter value k expresses some measure of the complexity of the graph
size of a minimum Vertex Cover,
P3 Cover,
Feedback Vertex Set,
etc.

13. Kernelization for graph problems
A kernelization algorithm takes (x, k) as input and computes an instance (x’, k’) of same problem in polynomial time, such that
Answer to x is YES  answer to x’ is YES
k’ ≤ k
|x’| ≤ f(k) for some function f
The function f is the size of the kernel
We want f to be a (small) polynomial
Kernelization reduces the size of the graph to something which depends
only on the complexity measure of the input,
not on the size of the input
Afterwards solve the smaller instances by some other method

14. Perspective for this talk
We want to solve the Independent Set problem
We use the solution values of the vertex deletion problems as complexity measures (parameters) of the input instances
Previous state of the art:
“Does graph G with vertex cover of size k have an independent set of size q?”
can be transformed in polynomial time into:
“Does graph G’ with vertex cover of size k’ have an independent set of size q’ ?”
where |G’| ≤ 2 k,
and k’ ≤ k.
Complexity-theoretic evidence that the factor 2 is optimal

15. Our results: upper bounds
“Does graph G with feedback vertex set of size k have an independent set of size q?”
can be transformed in polynomial time into:
“Does graph G’ with feedback vertex set of size k’ have an independent set of size q’ ?”
where |G’| ≤ O(k3),
and k’ ≤ k.
Our new bound uses more units of a smaller measure
|G’| ≤ O(|MinFVS|3)  |G’| ≤ 2 |MinVC|
Compare: “1000 ants weigh less than 3 horses”
Refined parameter
For simplicity we present the following result:
Transformation such that |G’| ≤ O(|MinP3Cover(G)|3).
The Independent Set problem parameterized by the size of a feedback vertex set admits a cubic-vertex kernel

16. Cubic-vertex kernel for parameter p3cover

17. Independent Set with P3-cover
Input: Graph G, modulator X such that G – X is a collection of paths on at most 2 vertices, integer q.
Question: Does G have an Independent Set of size q?
Parameter: k := |X|. X
G - X

18. Canonical solution structure
G - X
The maximum independent set (MIS) of G – X contains 1 vertex from each path in G – X
We call this a canonical solution for graph G
It uses no vertices of X
Poly-time computable
Vertices from X are only useful if they allow for a larger IS than the canonical solution X

19. Conflicts induced by a vertex in X
G - X
Consider vertex v in X
Compute a maximum independent set in G-X which avoids neighbors of v
Compare to the canonical solution (MIS in G-X)
Call the difference cf(v) the number of conflicts induced by v
Intuitively: the price we pay in G-X for using vertex v in an independent set
We can only improve on the canonical solution if the number of vertices we gain in X, is more than the number we lose in G-X X

20. Reduction rule 1 Deleting single vertices in X
G - X
If cf(v) ≥ |X| then delete v
There is always an optimal IS without v
Consider an IS using v
Might use |X| within X
Solution inside G-X at least |X| worse than canonical
Compare to:
Don’t use anything in X
Use optimum in G – X
(Canonical solution) X

21. Conflicts induced by pairs of vertices in X
G - X
Consider non-adjacent vertices {u,v} in X
Compute a maximum independent set in G-X which avoids neighbors of {u,v}
Compare to canonical solution
Call the difference cf({u,v}) the number of conflicts induced by{u,v}
Intuitively: the price we pay in G-X for using vertices {u,v} in an independent set X

22. Reduction rule 2 Adding edges in X
G - X
If cf({u,v})≥|X| then add edge {u,v}
There is always an optimal IS that avoids one of {u,v}
Consider an IS using {u,v}
Compared to the canonical solution it uses at least |X| less in G-X
So the canonical solution is at least as large
Does not use any vertices from X X

23. Reduction rule 3 Deleting P1 components from G-X
If there is an isolated vertex v in G – X which does not have any neighbors in X,
then delete v and decrease q by 1
We can always use v in an independent set
“Does G have an independent set of size q?” now reduces to “Does G – v have an independent set of size q-1?” X

24. Reduction rule 4 Deleting P2 components from G-X
If there is a P2 in G-X on vertices {x,y} such that
no single vertex in X sees {x,y},
no pair of non-adjacent vertices in X together sees {x,y}
then delete {x,y} and decrease q by 1
We can always use one of {x,y} in an independent set
No independent set in X contains neighbors of x and y simultaneously
“Does G have an independent set of size q?”
now reduces to
“Does G - {x,y} have an independent set of size q-1?” X
Observe:
P2’s in G – X that survive this rule have restricted structure!

25. Analysis
To prove: after exhausting these reduction rules we have |V| ≤ |X| + 2|X|2 + 2|X|3.
Count how many paths we have in G – X.
Type 1: All P1 and the P2 whose vertices have a common neighbor
Type 2: The P2 whose vertices have no common neighbor
Claim: # Type 1 ≤ |X|2.
Claim: # Type 2 ≤ |X|3.
G - X X

26. Type 1 ≤ |X|2
Type 1: All P1 and the P2 whose vertices have a common neighbor
A P1 path of Type 1 must be adjacent to a vertex in X (Rule 3)
A P2 path of Type 1 must be adjacent to a vertex in X (definition)
Claim: no vertex in X is adjacent to more than |X| paths of Type 1
If v in X is adjacent to |X| paths of Type 1, then cf(v) ≥ |X|
If we use v in IS, then we cannot use any vertices on adjacent Type 1 paths
IS size in G-X decreases by at least |X| if we use v
But by Rule 1 there are no vertices in X with cf(v) ≥ |X|
So we can charge all Type 1 paths to a (common) neighbor in X
We charge less than |X| to each vertex in X
Total charge = number of Type 1 paths ≤ |X|2 X
G - X

27. Type 2 ≤ |X|3
Type 2: The P2 whose vertices have no common neighbor
Claim: for Type 2 paths on {x,y} there are non-adjacent u,v in X such that u sees x, and v sees y
From definition of Type 2: no vertex in X sees both
If there is no such pair u,v then the path is deleted by Rule 4
Claim: if u,v in X are non-adjacent, then there are less than |X| P2’s in G – X such that u sees the left endpoint and v sees the right endpoint
If there are at least |X| such P2’s then cf({u,v}) ≥ |X| and we would add the edge {u,v} by Rule 2
So we can charge each P2 of Type 2 to some non-adjacent pair in X which sees the endpoints of this P2
We charge less than |X| to each pair
Total charge = number of Type 2 paths ≤ |X|2 • |X| = |X|3 X
G - X X
G - X

28. Summing it up
To prove: after exhausting these reduction rules we have |V| ≤ O(|X|3).
We proved bounds on the number of paths:
# Type 1 ≤ |X|2.
# Type 2 ≤ |X|3.
Number of vertices on a path is at most 2
Besides vertices on paths, graph contains only X.
|V|=|X| + |V(Type 1)| + |V(Type 2)| ≤ |X|+2(|X|2+|X|3).
Reduction rules can be applied in polynomial time
What is left of X forms a P3 Cover for the resulting graph
Complexity of final instance is not greater than of input instance

29. Cubic-vertex kernel for parameter FEEDBACK VERTEX SET
A sketch of the general result
Cubic-vertex kernel for parameter FEEDBACK VERTEX SET

30. Independent Set with Feedback Vertex Set
Input: Graph G, modulator X such that G – X is a forest, integer q.
Question: Does G have an Independent Set of size q?
Parameter: k := |X|.
Solve in 2|X|(|V| + |E|) time
Try all subsets S of X
Skip if S is not independent
Otherwise compute MIS in G-X which avoids neighbors of S
Solve MIS in G – X – N(S)
This is a forest!
Return maximum value of |S| + MIS
G - X X

31. Outline
We can still compute a canonical solution (MIS of G – X) in polynomial time since G – X is a forest
As before, number of conflicts induced by vertex v in X, or a non-adjacent pair {u,v} in X, is the decrease in the size of the solution within G – X, when using those vertices
Rule 1: Delete v in G – X with cf(v) ≥ |X|
Rule 2: Add edge between non-adjacent u,v in X if cf({u,v}) ≥ |X|
Rule 3: Delete a tree T in G – X if there are no non-adjacent vertices {u,v} in X which induce a conflict on T
Decrease q by MIS(T)
Not obvious that checking for pairs is enough
Rule 4, 5: Simplify structure of trees in G – X
Analysis:
charge vertices in a tree to neighbors in X
total charge cannot be too big without triggering reduction rules
20 pages of proof for the analysis

32. The modulator X in the input
We have assumed that we get the modulator X (the deletion set) as part of the input
Might not be the case in practice
Kernelization claims do not rely on X being a minimum set; the size of the reduced instance is bounded in |X|
So we compute a 2-approximation X, use it instead
|G’| is bounded in O(|X|3)
|X| is bounded by 2 |MinFVS(G)|
Hence |G’| is bounded by O(|MinFVS(G)|3)

33. no polynomial kernel for parameter p3cover
The weighted variant of the problem
no polynomial kernel for parameter p3cover

34. Weighted Independent Set with P3-cover
Input: Vertex-weighted graph G, modulator X such that G – X is a collection of paths on at most 2 vertices, integer q.
Question: Does G have an Independent Set of total weight at least q?
Parameter: k := |X|.
G - X X
Weight 12
Weight 30

35. Contrasting result
Weighted Independent Set with P3-cover does not admit a polynomial kernel
(assuming a widely-believed conjecture from complexity theory)
Proof uses a variation of many-one reductions
Intuition:
There is no answer-preserving polynomial-time procedure that reduces an instance of Weighted Independent Set to some instance whose size is bounded by the size of a P3 cover
Independent Set parameterized by P3 cover is the first example where the use of vertex weights does not affect fixed-parameter tractability, but does affect kernelizability
Compare: for Independent Set with parameter Vertex Cover both the weighted and unweighted problem admit small kernels!

36. Why vertex weights make the problem harder to kernelize
Main idea:
Build a graph G which contains adjacent pairs of vertices inside the modulator X
If you select exactly one from each pair, then the rest of the independent set behaves in some nice way
But any maximum cardinality independent set would not use any vertices from X at all
Give the vertices in these pairs high weight!
G - X X

37. conclusion and discussion

38. Summary of kernelization results
Independent Set
Weighted Independent Set
Parameter Vertex Cover
2k *
Parameter P3 Cover
O(k3)
No poly(k)
Parameter Feedback Vertex Set
Table shows number of vertices in reduced graphs
* marks existing results
Our results can be combined with existing kernelization
Ensures reduces instances using new technique are not bigger than using old technique

39. Kernelizability of (Unweighted) Independent Set
Feedback vtx Set
Forests
P3 cover
Paths ≤2 nodes
Vertex Cover
Edgeless Graphs ≤ ≤

40. Kernelizability of (Unweighted) Independent Set
Vertex Cover
Increasing size
P3 Cover
Cluster Deletion Distance
Feedback Vertex Set ? ?
Bipartite Deletion Distance
Outerplanar Deletion Distance
Treewidth

41. Kernel lower bounds for Unweighted Independent Set with structural parameters
Consider some graph class F such that
F is hereditary (closed under vertex deletion)
F contains all complete graphs
Maximum Independent Set can be solved in polynomial time for graphs in F
The independent set problem parameterized by the minimum number of vertices which have to be deleted to obtain a graph in class F, is in FPT
(assuming the deletion set S is given)
BUT: There is no polynomial kernel for this parameterized problem (unless …)
Proof using cross-composition

42. Implications
The Maximum Independent Set problem parameterized by the number k of vertices which have to be deleted to obtain a
Perfect graph,
Chordal graph,
Interval graph,
Cograph,
Etc …,
is in the class FPT but does not admit a polynomial kernel (unless …)

43. Conclusion
We have studied Independent Set parameterized by different measures of graph complexity
Size of a Vertex Cover, P3 Cover, Feedback Vertex Set
Usage of vertex weights affects kernelizability
Hierarchy of parameters (complexity measures) which we can explore
Open problems
Deletion distance to bipartite/outerplanar graphs
Improve the degree of the polynomial: cubic to quadratic?