From Approximative Kernelization to High Fidelity Reductions

joint with

Michael Fellows Ariel Kulik Frances Rosamond Technion Charles Darwin Univ.

Hadas ShachnaiTechnion

Workshop on Kernelization, Sept 2011

Approximative Kernelization• Traditionally: used as a preprocessing tool in FPT

algorithms, which does not harm the classification of the instance (as a ‘yes’ or ‘no’ w.r.t. the parameterized problem).

• Many FPT algorithms for NP-hard problems use kernels whose sizes are lower bounded by a function f(k) = Ω(poly(k)), where k is the parameter.

• Suppose that in solving an FPT problem Π, we want to obtain a kernel of smaller size (=better running time), with some compromise on its fidelity when lifting a solution for the kernelized instance back to a solution for the original instance.

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Can we define a tradeoff between fidelity and kernel size?

Approximative Kernelization Let L be a parameterized problem, i.e., L consists of

input pairs (x, k), where x is a problem instance, and k is the parameter. Given α ≥ 1, an α-fidelity kernelization of the problem

(i) Transforms in polynomial time the input (x, k) to ‘reduced’ input (x’, k’), such that k’ ≤ k and |x’| ≤ g(k, α), and

(ii) If (x, k) L then (x’, k’) L (iii) If (x’, k’) L then (x, α·k) L

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The special case where α = 1 is classic kernelization.

• Many 2- approximation polynomial-time algorithms

• Unless Unique Game Conjecture fails: No factor-(2- ε)-approximation polynomial time algorithm exists [Khot, Regev 2008].

• Vertex Cover is in FPT for general graphs: can be solved in time O*(1.28k).

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Application: Vertex CoverInput: An undirected graph G=(V,E), an integer k ≥ 1.Output: A subset of vertices C V, |C| ≤ k such that each

edge in E has at least one endpoint in C (if one exists).

Application: Vertex Cover

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1. Reduction step: Apply standard reduction rules to obtain (G’, k’).2. Shrinking step: Select l= ⌊k (α-1)⌋ independent edges, I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to I.3. Omit from G’ the subgraph induced by D; omit isolated vertices. Let G’’ be the resulting graph.4. The kernel is G’’ with k’’=k’- l.

Let G=(V,E), k ≥ 1 and α [1,2].

GGG

G’’ is α-fidelity kernel:1) G’’ is smaller than G’, therefore the size

requirement holds.2) If (G, k) L then (G’, k’) L (i.e., there is a cover C,such that |C| ≤ k’).

Application: Vertex Cover (Cont’d)

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1. Reduction step: Apply standard reduction rules to obtain (G’, k’).2. Shrinking step: Select l= k (α-1) independent edges, I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to I.3. Omit from G’ the subgraph induced by D; omit isolated vertices. Let G’’ be the resulting graph.4. The kernel is G’’ with k’’=k’- l.

GGG

Therefore, (G’’, k’’) L (C\D is a cover of size no greater than k’- l).3) If (G’’, k’’) L then (G, αk) L (there is a cover C’’ of size k’’ for G’’; then, C’’ U D is a cover of size k’’+2l =k’+lFor G’. Hence, there is a cover of size k+l = αk for G).

Kernel size is 2k(2- α).

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Algorithm : Shrinking step

v17

v14

v13

v19

v2

v4

v12

v18

v6

v3

v5

v7

v1

v8

v9

v11

v10

v16

v15

v20

l = k(α -1) =6

D={v1,v2,v4,v6,v7,v9,v12,v14,

v15,v16,v18,v19}

G’ = ({v1,…,v20}, E’)

k=10, α=8/5

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Algorithm : Shrinking step

v17

v13v3

v5

v8

v11

v10 v20

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Algorithm : Shrinking step

v5

v8

v11

v20

G’’= ({v5, v8,v11,v20}, E’’)

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Algorithm : example

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G=(V,E) , k=8

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Algorithm : example

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Reduction step:Omit the crownH1={b,c}I1={u,v,w}

α =2l = k(α -1) =8

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Algorithm : example

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Reduction step:Omit the crownH1={b,c}I1={u,v,w}

α =2

l = k(α-1) =8

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Algorithm : example

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Reduction step:Omit the crownH1={b,c}I1={u,v,w}

α =2

l = k(α -1) =8

|I| ‹ l : G’’ is a 2-fidelity kernel of size 0!

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Algorithm : example

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G=(V,E) , k=8

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Algorithm : example

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Reduction step:Omit the crownH1={b,c}I1={u,v,w}

α =1

l = k(α-1) =0

What Happens If We Switch the Order?

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1. Reduction step: Apply standard reduction rules to obtain

(G’, k’).2. Shrinking step: Select l= k (α-1) independent edges, I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to

I.3. Omit from G’ the subgraph induced by D; omit

isolated vertices. Let G’’ be the resulting graph.4. The kernel is G’’ with k’’=k’- l.

GGG

What Happens If We Switch the Order?

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1. Shrinking step: Select l= k (α-1) independent edges, I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to

I.2. Omit from G the subgraph induced by D; omit

isolated vertices. Let G’ be the resulting graph with k’=k- l.

3. Reduction step: Apply standard reduction rules to obtain

from G’ a kernel (G’’, k’’).

GGG

• G’’ is α-fidelity kernel.• Eliminates the assumption on linearity of

kernelization (we previously assumed that a cover of size k’+l for G’ implies a cover of size k+l for G).

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1. Shrinking step: Select l= k (α-1) independent edges,

I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to I.2. Omit from G the subgraph induced by D; omit isolated vertices. Let G’ be the resulting

graph with k’=k- l.3. Reduction step: Apply standard reduction

rules to obtain from G’ a kernel (G’’, k’’).

A parameterized Approximation Algorithm

Replace kernelization by any FPT algorithm.

Main contribution of Steps 1. and 2. is in decreasing the

value of k (tradeoff with fidelity).

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1. Shrinking step: Select l= k (α-1) independent edges,

I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to I.

2. Omit from G the subgraph induced by D; omit isolated vertices. Let G’ be the resulting graph with k’=k- l.

3. Solution step: Run FPT algorithm on G’.

A parameterized Approximation Algorithm

Let L be a parameterized problem, i.e., L consists of input pairs (x, k), where x is a problem instance, and k is the parameter. Given α ≥ 1, an α-fidelity shrinking of order h

(i) Transforms in polynomial time the input (x, k) to ‘reduced’ input (x’, k’), such that k’ ≤ h(k),

(ii) If (x, k) L then (x’, k’) L (iii) If (x’, k’) L then (x, α ·k) L

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A Generic Approach: α-fidelity Shrinking

Define simple reduction steps as a key building block to obtain α-fidelity shrinking for various problems

Given a parameterized problem L, a transformation r: U ─> U is (a,b)-reduction step if, for any (x, k) L,and (x’, k’) = r(x, k):

(i) k’ = k - a, (ii) If (x, k) L then (x’, k’) L (iii) For any integer n ≥ 0, if (x’, k’+n) L then

(x, k+b+n) L

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Obtaining α-fidelity Shrinking

Given a parameterized problem L, an (a,b)-reduction step r, and α ≤ 1+b/a, such that r can be evaluated in polynomial time, there is α-fidelity shrinking of orderk(b+ a - αa)/b for L.

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Approximations via α-fidelity Shrinking: Some Examples

Problem Kernel size Running time

Best FPT algorithm

Vertex cover 2(2-α)k 1.273 (2-α)k 1.273k

[CKX’06]

Connected vertex cover

No poly- kernel 2k(2-α)

2k [CN+ 11]

3-Hitting set

2.076k [W’07]

(All problems parameterized by solution size, k)

•

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How Powerful Is The Approach?

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Related Work FPT approximation

• Obtain a solution of value g(k) for a problem parameterized by k (e.g., Downey, F, McCartin and R, 2008; Many more..)

Parameterized approximations for NP-hard problems by moderately exponential time algorithms

• Improve best known approximation ratios for subgraph maximization, minimum covering (Bourgeois, Escoffier and Paschos, 2009)

• β-approximation algorithms for vertex cover, β(1,2), through accelerated branching (Fernau, Brankovic and Cakic, 2009)

• β-approximation algorithms for Hitting sets (Fernau, 2011)

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Related Work (Cont’d)

Links between approximation and kernelization:

• Exploit polynomial time approximation results in kernelization (Bevern, Moser and Niedermeier, 2010)

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What’s next?

Explore further the Generic approach for approximations based on α-fidelity shrinking: efficient application for other problems (e.g. Feedback Vertex Set, Edge Dominating Set..) ?

Can non-linear reduction (kernelization) rules be used to obtain better order (i.e., decreased running time)?

Extend the approach to problems with no FPT algorithm.