Specialization of
Kernelization
Generalization and
Daniel Lokshtanov
We
∃
Kernels
¬
Why?
Kernels
What’s Wrong with Kernels
(from a practitioners point of view)
1.
2.
3.
4.
Only handles NP-hard problems.
Don’t combine well with heuristics.
Only capture size reduction.
Don’t analyze lossy compression.
Doing something about (1) is a different field altogether.
Some preliminary work on (4)  high fidelity redections
This talk; attacking (2)
”Kernels don’t combine with
heuristics” ??
Kernel mantra; ”Never hurts to kernelize first,
don’t lose anything”
We don’t lose anything if after kernelization we
will solve the compressed instance exactly. Do
not necessarily preserve approximate
solutions.
Kernel
In this talk, parameter = solution size / quality
I,k
I’,k’
Solution of size ≤ k
Solution of size 1.2k
Solution of size ≤ k’
??
Solution of size 1.2k’
Known/Unknown k
Don’t know OPT in advance.

Solutions:
- The parameter k is given and we only care
whether OPT ≤ k or not.
Overhead
- Try all values for k.
- Compute k ≈ OPT by approximation algorithm.
If k > OPT, does kernelizing with k preserve OPT?
Buss kernel for Vertex Cover
Vertex Cover: Find S ⊆ V(G) of size ≤ k such that
every
edge
an endpoint
S. of correctness
Reduction
ruleshas
are independent
of k.in
Proof
transforms any solution, not only any optimal solution.
- Remove isolated vertices
- Pick neighbours of degree 1 vertices into
solution (and remove them)
- Pick degree > k vertices into solution and
remove them.
Degree > k rule
Any solution of size ≤ k must contain all vertices
of degree > k.
We preserve all solutions of size ≤ k.
Lose information about solutions of size ≥ k.
Buss’ kernel for Vertex Cover
- Find a 2-approximate solution S.
- Run Buss kernelization with k = |S|.
I,k
Solution of size
1.2k’ + (k-k’) ≤
1.2k
I,k’
Solution of size 1.2k’
Buss’ - kernel
- Same size as Buss kernel, O(k2), up to
constants.
- Preserves approximate solutions, with no loss
compared to the optimum in the compression
and decompression steps.
NT-Kernel
In fact the Nemhauser Trotter 2k-size kernel for
vertex cover already has this property – the
crown reduction rule is k-independent!
Proof: Exercise 
Other problems
For many problems applying the rules with a
value of k preserves all ”nice” solutions of size
≤ k  approximation preserving kernels.
Example 2: Feedback Vertex Set, we adapt a
O(k2) kernel of [T09].
Feedback Vertex Set
FVS: Is there a subset S ⊆ V(G) of size ≤ k such
that G \ S is acyclic?
R1 & R2 preserve all reasonable solutions
R1: Delete vertices of degree 0 and 1.
R2: Replace degree 2 vertices by edges.
R3 preserves all solutions of size ≤ k
R3: If v appears in > k cycles that intersect only
in v, select v into S.
Feedback Vertex Set
R4 (handwave): If R1-R3 can’t be applied and
there is a vertex x of degree > 8k, we can
identify a set X such that in any feedback
vertex set S of size ≤ k, either x ∈ S or X ⊆ S.
R4 preserves all solutions of size ≤ k
Feedback Vertex Set Kernel
Apply a 2-approximation algorithm for Feedback
Vertex Set to find a set S.
Apply the kernel with k=|S|. Kernel size is
O(OPT2).
Preserves approximate solutions, with no loss
compared to the optimum in the compression
step.
Remarks;
If we don’t know OPT, need an approximation
algorithm.
Most problems that have polynomial kernels
also have constant factor or at least Poly(OPT)
approximations.
Using f(opt)-approximations to set k results in
larger kernel sizes for the approximation
preserving kernels.
Right definition?
Approximation preserving kernels for
optimization problems, definition 1:
OPT
I
Poly time
OPT’
I’
|I’I≤ poly(OPT)
Poly time
c*OPT
c*OPT’
Right definition?
Approximation preserving kernels for
optimization problems, definition 2:
OPT
I
Poly time
OPT’
I’
|I’I≤ poly(OPT)
Poly time
OPT + t
OPT’ + t
What is the right definition?
Definition 1 captures more, but Definition 2
seems to capture most (all?) positive answers.
Exist other reasonable variants that are not
necessarily equivalent.
What do approximation preserving
kernels give you?
When do approximation preserving kernels help
in terms of provable running times?
If Π has a PTAS or EPTAS, and an approximation
preserving kernel, we get (E)PTASes with
running time f(ε)poly(OPT) + poly(n) or
OPTf(ε) + poly(n).
Problems on planar (minor-free)
graphs
Many problems on planar graphs and H-minorfree graphs admit EPTAS’s and have linear
kernels.
Make the kernels approximation preserving?
These Kernels have only one reduction
rule; the protrusion rule.
(to rule them all)
Protrusions
A set S ⊆ V(G) is an r-protrusion if
- At most r vertices in S have neighbours
outside S.
- The treewidth of G[S] is at most r.
Protrusion Rule
A protrusion rule takes a graph G with an rprotrusion S of size > c, and outputs an
equivalent instance G’, with V(G’) < V(G).
The constant c depends on the problem
and on r.
Usually, the entire part G[S] is replaced by a
different and smaller protrusion that
”emulates” the behaviour of S.
Kernels on Planar Graphs
[BFLPST09]: For many problems, a protrusion
rule is sufficient to give a linear kernel on
planar graphs.
To make these kernels apx-preserving, we need
an apx-preserving protrusion rule.
Apx-Preserving Protrusion Rule
S
OPT
I
Poly time
I’
|I’I< I
Poly time
OPT + t
OPT’ + t
OPT’≤
OPT
Kernels on Planar Graphs
[BFLPST09]:
– If a problem has finite integer index  it has a
protrusion rule.
– Simple to check sufficient condition for a problem
to have finite integer index.
Finite integer index is not enough for apxpreserving protrusion rule. But the sufficient
condition is!
t-boundaried graphs
A t-boundaried graph is a graph G with t
distinguished vertices labelled from 1 to t.
These vertices are called the boundary of G.
G can be colored, i.e supplied with some
vertex/edge sets C1,C2…
C1
C2
Gluing
Gluing two colored t-boundaried graphs:
(G1,C1,C2) ⊕ (G2,D1,D2)  (G3, C1∪ D1, C2∪ D2)
means identifying the boundary vertices with
the same label, vertices keep their colors.
C1
1
C1
C2
C2
1
1
2
3
2
3
2
3
D2 D2 D
D1 1
Canonical Equivalence
For a property Φ of 1-colored graphs we define
the equivalence relation ≣Φ on the set of tboundaried c-colored graphs.
(G1,X1) ≣Φ (G2,X2) ⇔ For every (G’, X’):
Φ(G1⊕G’, X1∪X’) ⇔ Φ(G2⊕G’, X2∪ X’)
Can also define for 10-colorable problems in the same way
Canonical Equivalence
(G1,X) ≣Φ (G2,Y) means ”gluing (G1,X) onto
something has the same effect as gluing (G2,Y)
onto it”
X1
1
2
X2
3
1
2
3
Y1
Y2
1
2
3
Z2
Z1
Finite State
Φ is finite state if for every integer t, ≣Φ has a
finite number of equivalence classes on tboundaried graphs.
Note: The number of equivalence classes is a
function f(Φ,t) of Φ and t.
Variant of Courcelle’s Theorem
Finite State Theorem (FST): If Φ is CMSOLdefinable, then Φ is finite state.
Quantifiers: ∃ and ∀ for variables for vertex sets
and edge sets, vertices and edges.
Operators: = and ∊
Operators: inc(v,e) and adj(u,v)
Logical operators: ∧, ∨ and ¬
Size modulo fixed integers operator: eqmodp,q(S)
EXAMPLE: p(G,S) = “S is an independent set of G”:
p(G,S) = ∀u, v ∊ S, ¬adj(u,v)
CMSOL Optimization Problems
for colored graphs
Φ-Optimization
Input: G, C1, ... Cx
Max / Min |S|
So that Φ(G, C1, Cx, S) holds.
CMSOL definable proposition
Sufficient Condition
[BFLPST09]:
– If a CMSO-optimization problem Π is strongly
monotone  Π has finite integer index  it has a
protrusion rule.
Here:
– If a CMSO-optimization problem Π is strongly
monotone  Π has apx-preserving protrusion
rule.
Signatures
(for minimization problems)
Choose smallest S ⊆ V(G) to make Φ hold
SH1
H1
2
|SG2|=5
|SG1| = 2
G |S
G3|=1
Intuition: f(H,S) returns the best way to
complete in G a fixed partial solution in H.
SH2
SH3
H2
5
H3
1
Signatures
(for minimization problems)
The signature of a t-boundaried graph G is a
function fG with
Input: t-boundaried graph H and SH ⊆ V(H)
Output: Size of the smallest SG ⊆ V(G) such that
Φ(G ⊕ H, SG ∪ SH) holds.
Output: ∞ if SG does not exist.
Strong Monotonicity
(for minimization problems)
A problem Π is strongly monotone if for any tboundaried G, there is a vertex set Z ⊆ V(G)
such that |Z| ≤ fG(H,S) + g(t) for an arbitrary
function g.
Signature of G, evaluated at (H,S)
Size of the smallest S’ ⊆ V(G) such that S’ ∪ S
is a feasible solution of G ⊕ H
Strong monotonicity - intuition
Intuition: A problem is strongly monotone if for
any t-boundaried G there ∃ partial solution S
that can be glued onto ”anything”, and S is
only g(t) larger than the smallest partial
solution in G.
Super Strong Monotonicity Theorem
Theorem: If a CMSO-optimization problem Π is
strongly monotone, then it has apx-preserving
protrusion rule.
Corollary: All bidimensional’, strongly monotone
CMSO-optimization problems Π have linear
size apx-preserving kernels on planar graphs.
Proof of SSMT
Lemma 1: Let G1 and G2 be t-boundaried graphs of
constant treewidth, f1 and f2 be the signatures of
G1 and G2, and c be an integer such that for any
H, SH ⊆ V(H): f1(H,SH) + c = f2(H,SH). Then:
Decrease size by c
Feasible solution
Z1
G1 ⊕ H
Poly time
Increase size by c
Poly time
Feasible solution
Z2
G2 ⊕ H
Proof of Lemma 1
G1
Decrease size by c
H
Poly time?
G2
Constant treewidth!
H
Proof of SSMT
Lemma 2: If a CMSO-min problem Π is strongly
monotone, then:
For every t there exists a finite collection F of tboundaried graphs such that:
For every G1, there is a G2 ∈ F and c ≥ 0 such that:
For any H, SH ⊆ V(H): f1(H,SH) + c = f2(H,SH).
SSMT = Lemma 1 + 2
Keep a list F of graphs t-boundaried graphs as
guaranteed by Lemma 2.
Replace large protrusions by the corresponding
guy in F. Lemma 1 gives correctness.
Proof of Lemma 2
Signature
value
G2
G1
(H1, S1) (H2, S2) (H3, S3)(H4, S4)(H5, S5)(H6, S6)(H7, S7)(H8, S8)...
≤ g(t)
Proof of Lemma 2
Only a constant number of finite, integer curves
that satisfy max-min ≤ t (up to translation).
Infinite number of infinite such curves.
Since Π is a min-CMSO problem, we only need
to consider the signature of G on a finite
number of pairs (Hi,Si).
Super Strong Monotonicity Theorem
Theorem: If a CMSO-optimization problem Π is
strongly monotone, then it has apx-preserving
protrusion rule.
Corollary: All bidimensional’, strongly monotone
CMSO-optimization problems Π have linear
size apx-preserving kernels on planar graphs.
Recap
Approximation preserving kernels are much
closer to the kernelization ”no loss” mantra.
It looks like most kernels can be made
approximation preserving at a small cost.
Is it possible to prove that some problems have
smaller kernels than apx-preserving kernels?
What I was planning to talk about, but
didn’t.
”Kernels” that do not reduce size, but rather
reduce a parameter to a function of another in
polynomial time.
– This IS pre-processing
– Many many examples exist already
– Fits well into Mike’s ”multivariate” universe.