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Generalization and Specialization of Kernelization Daniel Lokshtanov

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We 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(k 2 ), 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(k 2 ) 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(OPT 2 ). 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 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 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 OPT f(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 problem and 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 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 C 1,C 2 … C1C1 C2C2

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Gluing Gluing two colored t-boundaried graphs: (G 1,C 1,C 2 ) ⊕ (G 2,D 1,D 2 ) (G 3, C 1 ∪ D 1, C 2 ∪ D 2 ) means identifying the boundary vertices with the same label, vertices keep their colors. C1C1 C2C D2D2 D1D C1C1 C2C2 D2D2 D1D

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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. (G 1,X 1 ) ≣ Φ (G 2,X 2 ) ⇔ For every (G’, X’): Φ(G 1 ⊕ G’, X 1 ∪ X’) ⇔ Φ(G 2 ⊕ G’, X 2 ∪ X’) Can also define for 10-colorable problems in the same way

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Canonical Equivalence (G 1,X) ≣ Φ (G 2,Y) means ”gluing (G 1,X) onto something has the same effect as gluing (G 2,Y) onto it” X1X1 X2X Z2Z2 Z1Z Y1Y1 Y2Y

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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 Theorem Finite 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: eqmod p,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 Problems for colored graphs Φ-Optimization Input: G, C 1,... C x Max / Min |S| So that Φ(G, C 1, C x, 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 H3H3 H2H2 H1H1 S H3 S H2 S H1 |S G1 | = 2 |S G3 |=1 |S G2 |= 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 f G with Input: t-boundaried graph H and S H ⊆ V(H) Output: Size of the smallest S G ⊆ V(G) such that Φ(G ⊕ H, S G ∪ S H ) holds. Output: ∞ if S G 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| ≤ f G (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 G 1 and G 2 be t-boundaried graphs of constant treewidth, f 1 and f 2 be the signatures of G 1 and G 2, and c be an integer such that for any H, S H ⊆ V(H): f 1 (H,S H ) + c = f 2 (H,S H ). Then: G 1 ⊕ H Feasible solution Z 1 G 2 ⊕ H Feasible solution Z 2 Poly time Decrease size by c Poly time Increase size by c

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Proof of Lemma 1 G1G1 H H G2G2 Decrease size by c Poly 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 G 1, there is a G 2 ∈ F and c ≥ 0 such that: For any H, S H ⊆ V(H): f 1 (H,S H ) + c = f 2 (H,S H ).

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SSMT = Lemma 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 (H 1, S 1 ) Signature value (H 2, S 2 )(H 3, S 3 )(H 4, S 4 )(H 5, S 5 )(H 6, S 6 )(H 7, S 7 )(H 8, S 8 )... G1G1 ≤ g(t) G2G2

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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 (H i,S i ).

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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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