generalization and specialization of kernelization
DESCRIPTION
Generalization and Specialization of Kernelization. Daniel Lokshtanov. We. Kernels. ¬. ∃. Kernels. Why?. What’s Wrong with Kernels (from a practitioners point of view). Only handles NP-hard problems. Don’t combine well with heuristics . Only capture size reduction . - PowerPoint PPT PresentationTRANSCRIPT
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Generalization and Specialization of Kernelization
Daniel Lokshtanov
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We Kernels
∃ ¬ Kernels
Why?
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What’s Wrong with Kernels(from a practitioners point of view)
1. Only handles NP-hard problems.2. Don’t combine well with heuristics.3. Only capture size reduction.4. Don’t analyze lossy compression.
Doing something about (1) is a different field altogether.
This talk; attacking (2)
Some preliminary work on (4) high fidelity redections
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”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.
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Kernel
I,k I’,k’
In this talk, parameter = solution size / quality
Solution of size ≤ k Solution of size ≤ k’
Solution of size 1.2k’Solution of size 1.2k??
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Known/Unknown k
Don’t know OPT in advance.
Solutions:- The parameter k is given and we only care
whether OPT ≤ k or not. - Try all values for k.- Compute k ≈ OPT by approximation algorithm.
Overhead
If k > OPT, does kernelizing with k preserve OPT?
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Buss kernel for Vertex Cover
Vertex Cover: Find S V(G)⊆ of size ≤ k such that every edge has an endpoint in S.
- Remove isolated vertices- Pick neighbours of degree 1 vertices into
solution (and remove them)- Pick degree > k vertices into solution and
remove them.
Reduction rules are independent of k. Proof of correctness transforms any solution, not only any optimal solution.
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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.
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Buss’ kernel for Vertex Cover
- Find a 2-approximate solution S.- Run Buss kernelization with k = |S|.
I,k I,k’
Solution of size 1.2k’Solution of size 1.2k’ + (k-k’) ≤1.2k
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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.
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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
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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].
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Feedback Vertex Set
FVS: Is there a subset S V(G)⊆ of size ≤ k such that G \ S is acyclic?
R1: Delete vertices of degree 0 and 1.R2: Replace degree 2 vertices by edges.
R3: If v appears in > k cycles that intersect only in v, select v into S.
R1 & R2 preserve all reasonable solutions
R3 preserves all solutions of size ≤ k
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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
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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.
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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.
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Right definition?
Approximation preserving kernels for optimization problems, definition 1:
I I’|I’I≤ poly(OPT)
OPT
c*OPT
OPT’Poly time
Poly time
c*OPT’
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Right definition?
Approximation preserving kernels for optimization problems, definition 2:
I I’|I’I≤ poly(OPT)
OPT
OPT + t
OPT’Poly time
Poly time
OPT’ + t
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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.
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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).
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Problems on planar (minor-free) graphs
Many problems on planar graphs and H-minor-free 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)
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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.
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Protrusion Rule
A protrusion rule takes a graph G with an r-protrusion S of size > c, and outputs an equivalent instance G’, with V(G’) < V(G).
Usually, the entire part G[S] is replaced by a different and smaller protrusion that ”emulates” the behaviour of S.
The constant c depends on the problemand on r.
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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.
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Apx-Preserving Protrusion Rule
I I’|I’I< I
OPT
OPT + t
OPT’≤ OPT
Poly time
Poly time
OPT’ + t
S
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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 apx-preserving protrusion rule. But the sufficient condition is!
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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
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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 C2
12
3
D2 D1
12
3C1 C2
D2 D1
12
3
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Canonical Equivalence
For a property Φ of 1-colored graphs we define the equivalence relation ≣Φ on the set of t-boundaried c-colored graphs.
(G1,X1) ≣Φ (G2,X2) For every (G’, X’):⇔
Φ(G1 G’, X⊕ 1 X’) ∪ ⇔ Φ(G2 G’, X⊕ 2 X’) ∪ Can also define for 10-colorable problems in the same way
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Canonical Equivalence
(G1,X) ≣Φ (G2,Y) means ”gluing (G1,X) onto something has the same effect as gluing (G2,Y) onto it”
X1 X2
123
Z2 Z1
12
3
Y1
Y2 123
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Finite State
Φ is finite state if for every integer t, ≣Φ has a finite number of equivalence classes on t-boundaried graphs.
Note: The number of equivalence classes is a function f(Φ,t) of Φ and t.
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Variant of Courcelle’s TheoremFinite State Theorem (FST): If Φ is CMSOL-
definable, 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)∀ ∊
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CMSOL Optimization Problemsfor colored graphs
Φ-OptimizationInput: G, C1, ... Cx
Max / Min |S|So that Φ(G, C1, Cx, S) holds.
CMSOL definable proposition
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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.
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Signatures (for minimization problems)
G
H3
H2
H1
SH3
SH2
SH1
|SG1| = 2
|SG3|=1
|SG2|=5
2
5
1
Choose smallest S V(G) ⊆ to make Φ hold
Intuition: f(H,S) returns the best way to complete in G a fixed partial solution in H.
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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 S∪ H) holds.
Output: ∞ if SG does not exist.
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Strong Monotonicity(for minimization problems)
A problem Π is strongly monotone if for any t-boundaried 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⊕
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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.
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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.
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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:
G1 H⊕
Feasible solution
Z1
G2 H⊕
Feasible solution
Z2 Poly time
Decrease size by c
Poly time
Increase size by c
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Proof of Lemma 1
G1
H
H
G2
Decrease size by cPoly time?
Constant treewidth!
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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 t-boundaried 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).
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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.
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Proof of Lemma 2
(H1, S1)
Signaturevalue
(H2, S2) (H3, S3) (H4, S4)(H5, S5)(H6, S6)(H7, S7)(H8, S8)...
G1 ≤ g(t)
G2
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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).
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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.
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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?
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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.
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THANK YOU!