Bart Jansen
Independent Set Kernelization for a Refined
Parameter: Upper and Lower bounds
Joint work with Hans Bodlaender
1
TACO Day, Utrecht
January 12th, 2011
Independent Set Kernelization for a
Refined Parameter:
Upper and Lower bounds
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Introduction
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Independent Set
Parameters
Kernelization
Upper bounds
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Small kernel for parameter P3 cover
 Reduction rules
 Analysis
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Lower bounds
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2
Ideas for for parameter Feedback Vertex Set
Effect of introducing vertex weights
Conclusion
Our target problem
INDEPENDENT SET
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Independent Set
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Input:
Question:
Graph G, integer q
Is there a set S of ≥ q vertices which are
pairwise non-adjacent?
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4
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
Solutions to vertex deletion problems as complexity measures
PARAMETERS
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Vertex Deletion Problems
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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
Vertex Cover
Set of size ≥ Edgeless
n – q? Graphs
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Vertex Deletion Problems
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P3 Cover
Input:
Question:
Graph G, integer q
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
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Vertex Cover
Edgeless Graphs
Vertex Deletion Problems
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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
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P3 cover
Paths ≤2 nodes
Vertex Cover
Edgeless Graphs
Graph Complexity Measures

We can use the minimum sizes of these vertex deletion
sets as measures of the complexity of a graph
Feedback vtx Set
Forests
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Vertex Cover
Edgeless Graphs
Every edgeless graph is a collection of paths on ≤ 2 nodes
Every collection of paths on ≤ 2 nodes is a forest
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P3 cover
Paths ≤2 nodes
Difference between the parameters can be unbounded
Graph families
All graphs
Forests
Collections of
paths on ≤ 2
vertices
Edgeless graphs
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Attacking hard problems with small parameters
KERNELIZATION
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Graph problems
with structural parameters
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Consider a computational decision problem on graphs
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Parameter value k expresses some measure of the
complexity of the graph
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Input:
encoding x of a question about graph G, integer k.
Question: does graph G have a (…)?
Parameter:k
size of a minimum Vertex Cover,
P3 Cover,
Feedback Vertex Set,
etc.
Kernelization for graph problems
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A kernelization algorithm takes (x, k) as input and computes
an instance (x’, k’) of same problem in polynomial time, such
that
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The function f is the size of the kernel
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We want f to be a (small) polynomial
Kernelization reduces the size of the graph to something which
depends
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Answer to x is YES  answer to x’ is YES
k’ ≤ k
|x’| ≤ f(k) for some function f
only on the complexity measure of the input,
not on the size of the input
Afterwards solve the smaller instances by some other method
Perspective for this talk
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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
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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:
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“Does graph G’ with vertex cover of size k’ have an independent
set of size q’ ?”
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where
and
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|G’| ≤ 2 k,
k’ ≤ k.
Complexity-theoretic evidence that the factor 2 is optimal
Our results: upper bounds
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“Does graph G with feedback vertex set of size k have an independent
set of size q?”
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“Does graph G’ with feedback vertex set of size k’ have an independent
set of size q’ ?”
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where
and
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Our new bound uses more units of a smaller measure
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|G’| ≤ O(k3),
k’ ≤ k.
|G’| ≤ O(|MinFVS|3)  |G’| ≤ 2 |MinVC|
Refined parameter
For simplicity we present the following result:
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can be transformed in polynomial time into:
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
CUBIC-VERTEX KERNEL
FOR PARAMETER P3COVER
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Independent Set with P3-cover
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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|.
G-X
X
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Canonical solution structure
 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
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 Vertices from X are only
useful if they allow for a
larger IS than the
canonical solution
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G-X
X
Conflicts induced by a vertex in X
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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
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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
G-X
X
Reduction rule 1
Deleting single vertices in X
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If cf(v) ≥ |X| then delete v
 There is always an optimal
IS without v
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Consider an IS using v
 Might use |X| within X
 Solution inside G-X at least
|X| worse than canonical
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Compare to:
 Don’t use anything in X
 Use optimum in G – X
(Canonical solution)
G-X
X
Conflicts induced by pairs of vertices in 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}
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Intuitively: the price we
pay in G-X for using
vertices {u,v} in an
independent set
G-X
X
Reduction rule 2
Adding edges in X
 If cf({u,v})≥|X| then
add edge {u,v}
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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
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G-X
X
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
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“Does G have an
independent set of size q?”
now reduces to
“Does G – v have an
independent set of size q-1?”
G-X
X
Reduction rule 4
Deleting P2 components from G-X
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no single vertex in X sees {x,y},
no pair of non-adjacent vertices
in X together sees {x,y}
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then delete {x,y} and decrease
q by 1
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We can always
use one of {x,y} in
Observe:
an independent set
P2’s in
X that set
survive
this rule
 G
No–
independent
in X contains
neighbors of x and y simultaneously
have restricted structure!

“Does G have an independent set of
size q?”
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G-X
If there is a P2 in G-X on vertices
{x,y} such that both
now reduces to
“Does G - {x,y} have an independent
set of size q-1?”
X
Analysis
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After exhausting the reduction rules:
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each single vertex induces at most |X|
conflicts
each non-adjacent pair induces at most |X|
conflicts
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Total number of conflicts at most |X|2 + |X|3
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Not hard to show that each path in G – X
contributes to the number of induced
conflicts
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# vertices per path is ≤ 2
# vertices in G – X is ≤ 2(|X|2 + |X|3)

|V| ≤ |X| + 2(|X|2 + |X|3) = O(|X|3)
G-X
X
Summing it up
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Reduction rules can be applied in polynomial time
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What is left of X forms a P3 Cover for the resulting graph
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Complexity of final instance is not greater than of input
instance
Independent Set parameterized by the size of a P3 Cover
admits a kernel with O(k3) vertices
A sketch of the general result
CUBIC-VERTEX KERNEL FOR
PARAMETER FEEDBACK VERTEX SET
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Independent Set with Feedback Vertex Set
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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
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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)
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This is a forest!
Return maximum value of |S| + MIS
G-X
X
Outline
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We can still compute a canonical solution (MIS of G – X) in polynomial
time since G – X is a forest
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As before, number of conflicts induced by vertex v in X, or a nonadjacent pair {u,v} in X, is the decrease in the size of the solution
within G – X, when using those vertices
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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
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Rule 4, 5: Simplify structure of trees in G – X
Analysis:
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Decrease q by MIS(T)
Not obvious that checking for pairs is enough
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
The modulator X in the input
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We have assumed that we get the modulator X (the
deletion set) as part of the input
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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
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Might not be the case in practice
|G’| is bounded in O(|X|3)
|X| is bounded by 2 |MinFVS(G)|
Hence |G’| is bounded by O(|MinFVS(G)|3)
The weighted variant of the problem
NO POLYNOMIAL KERNEL
FOR PARAMETER P3COVER
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Weighted Independent Set with P3-cover
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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
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Contrasting result
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Weighted Independent Set with P3-cover does not admit a
polynomial kernel
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Intuition:
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(assuming a widely-believed conjecture from complexity theory)
Proof uses a variation of many-one reductions
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
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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
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Compare: for Independent Set with parameter Vertex Cover
both the weighted and unweighted problem admit small
kernels!
Why vertex weights make the problem
harder to kernelize
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Main idea:
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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!
X
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G-X
CONCLUSION AND
DISCUSSION
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Summary of kernelization results
Independent Set
Weighted
Independent Set
Parameter
Vertex
Cover
2k *
2k *
Parameter
P3 Cover
O(k3)
No poly(k)
Parameter
Feedback
Vertex Set
O(k3)
No poly(k)
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Table shows number of vertices in reduced graphs
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Our results can be combined with existing kernelization
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36
* marks existing results
Ensures reduces instances using new technique are not bigger than
using old technique
Kernelizability of (Unweighted)
Independent Set
Feedback vtx Set
Forests
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P3 cover
Paths ≤2 nodes
Vertex Cover
Edgeless Graphs
Kernelizability of (Unweighted)
Independent Set
Increasing size
Vertex
Cover
P3 Cover
Clique
Deletion
Distance
Feedback
Vertex Set
Bipartite
Deletion
Distance
Outerplanar
Deletion
Distance
Treewidth
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Kernel lower bounds for Unweighted Independent
Set with structural parameters
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Consider some graph class F such that
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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
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(assuming the deletion set X is given)
BUT: There is no polynomial kernel for this parameterized
problem (unless …)
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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
Proof using cross-composition
[With Hans Bodlaender and Stefan Kratsch]
Implications

The Maximum Independent Set problem parameterized by
the number k of vertices which have to be deleted to
obtain a
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Perfect graph,
Chordal graph,
Interval graph,
Cograph,
Etc …,
is in the class FPT but does not admit a polynomial kernel
(unless …)
Conclusion
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We have studied Independent Set parameterized by
different measures of graph complexity
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Usage of vertex weights affects kernelizability
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Hierarchy of parameters (complexity measures) which we
can explore
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Open problems
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Size of a Vertex Cover, P3 Cover, Feedback Vertex Set
Deletion distance to bipartite/outerplanar graphs
Improve the degree of the polynomial: cubic to quadratic?